# On Fourier coefficients and Hecke eigenvalues of Siegel cusp forms of degree
$2$

Biplab Paul and Abhishek Saha  School of Mathematical Sciences, Queen Mary
University of London, Mile End Road, London, E1 4NS, UK.
Current Address: Chennai Mathematical Institute, H1, SIPCOT IT Park, Siruseri,
Kelambakkam 603103, Tamil Nadu, India<EMAIL_ADDRESS><EMAIL_ADDRESS>School of Mathematical Sciences, Queen Mary University of London, Mile End
Road, London, E1 4NS, UK<EMAIL_ADDRESS>
###### Abstract.

We investigate some key analytic properties of Fourier coefficients and Hecke
eigenvalues attached to scalar-valued Siegel cusp forms $F$ of degree 2,
weight $k$ and level $N$. First, assuming that $F$ is a Hecke eigenform that
is not of Saito-Kurokawa type, we prove an improved bound in the $k$-aspect
for the smallest prime at which its Hecke eigenvalue is negative. Secondly, we
show that there are infinitely many sign changes among the Hecke eigenvalues
of $F$ at primes lying in an arithmetic progression. Third, we show that there
are infinitely many positive as well as infinitely many negative Fourier
coefficients in any “radial” sequence comprising of prime multiples of a fixed
fundamental matrix. Finally we consider the case when $F$ is of Saito–Kurokawa
type, and in this case we prove the (essentially sharp) bound
$|a(T)|~{}\ll_{F,\epsilon}~{}\big{(}\det T\big{)}^{\frac{k-1}{2}+\epsilon}$
for the Fourier coefficients of $F$ whenever $\gcd(4\det(T),N)$ is squarefree,
confirming a conjecture made (in the case $N=1$) by Das and Kohnen.

###### Key words and phrases:

Siegel cusp forms, Hecke eigenvalues, Fourier coefficients, elliptic modular
forms

###### 2010 Mathematics Subject Classification:

11F30, 11F46

##  1\. Introduction

For positive integers $n,k$, and $N$, let $S_{k}(\Gamma^{(n)}_{0}(N))$ be the
space of holomorphic cusp forms of weight $k$ for the Siegel-type congruence
subgroup $\Gamma^{(n)}_{0}(N)\subseteq\mathrm{Sp}_{2n}({\mathbb{Z}})$ of level
$N$ and degree $n$. Any $F\in S_{k}(\Gamma^{(n)}_{0}(N))$ has a Fourier
expansion

$F(Z)~{}=~{}\sum_{T\in{\mathcal{N}}_{n}}a(T)e^{2\pi i\rm{tr}(TZ)},$

where

${\mathcal{N}}_{n}~{}:=~{}\big{\\{}T=(t_{ij})_{n\times
n}~{}|~{}2t_{ij},t_{ii}\in{\mathbb{Z}},~{}T^{t}=T,~{}T>0\big{\\}}.$

Here and throughout the article, the symbol $T>0$ means $T$ is positive
definite. If $F$ is a Hecke eigenform for the Hecke operators $T(m)$ with
$\gcd(m,N)=1$, we denote its Hecke eigenvalues by $\eta(m)$. Both the Fourier
coefficients $a(T)$ and Hecke eigenvalues $\eta(m)$ are key objects and they,
separately and together, play an important role in understanding the Hecke
eigenforms and hence all cusp forms.

It is well-known that when $n=1$, i.e., when $F$ is an elliptic Hecke
eigenform, the Fourier coefficients and Hecke eigenvalues coincide up to a
certain normalization. In this case, one can exploit the multiplicative
properties of $\eta(m)$, the Ramanujan bound, and the Hecke relations to study
sign change questions for $\eta(m)$. Some interesting results in this set-up
can be found in [KS06, KLSW10, Mat12, MRl15, GKP19, GMP21] and the references
therein.

In the case of Siegel cusp forms of degree $n=2$, Hecke eigenvalues and
Fourier coefficients of a Hecke eigenform are not related in any simple way.
Further, it is not clear how to make sense of multiplicative properties of
Fourier coefficients (which are indexed by matrices). Indeed, Fourier
coefficients are closely related to central values of degree 8 $L$-functions
and are more mysterious than the Hecke eigenvalues [DPSS20, JLS21, FM21]. For
example, even though an analogue of Ramanujan’s conjecture is known for the
Hecke eigenvalues corresponding to an eigenform of degree $n=2$, the analogous
conjecture for Fourier coefficients (which is famously known as
Resnikoff–Saldaña conjecture [RSn74]) is not known yet. For recent progress
towards the Resnikoff–Saldaña conjecture, see [JLS21]. In this paper we
investigate the following questions for $n=2$:

  * •

Bounds for the first prime for which the Hecke eigenvalue is negative
(analogue of least quadratic non-residue).

  * •

Sign changes of Hecke eigenvalues at primes lying in any fixed arithmetic
progression.

  * •

Behaviour of signs of Fourier coefficients of a non-zero cusp form (not
necessarily a Hecke eigenform) along a sequence of prime multiples of a fixed
matrix.

  * •

Bounds for the Fourier coefficients of a cusp form lying in the generalized
Maass subspace.

Throughout this paper, we allow the level $N$ to be any integer, in contrast
to many previous related works which were restricted to $N=1$ or $N$
squarefree.

###  1.1. Main Results on signs of Hecke eigenvalues

In the case $N=1$, the space $S_{k}({\rm
Sp}_{4}({\mathbb{Z}}))=S_{k}({\Gamma}_{0}^{(2)}(1))$ of Siegel cusp forms of
weight $k$ and full level has a natural subspace denoted $S_{k}^{*}({\rm
Sp}_{4}({\mathbb{Z}}))$ and called the _Maass subspace_ , which is the span of
the _Saito-Kurokawa lifts_ of full level. The Saito-Kurokawa lifts of full
level can be explicitly constructed from classical half-integral weight forms
via the theory of Jacobi forms [EZ85, §6]; they may also be viewed as lifts of
classical integral weight forms thanks to the Shimura correspondence between
half integral weight and integral weight forms. It is known [Bre99, GPS18]
that all the Hecke eigenvalues of a Hecke eigenform lying in the Maass
subspace are positive. Indeed, for a Hecke eigenform $F$ lying in the Maass
subspace, there exists a classical Hecke eigenform $f\in S_{2k-2}({\rm
SL}_{2}({\mathbb{Z}}))$ such that

(1) $\lambda_{F}(p)=\lambda_{f}(p)+p^{1/2}+p^{-1/2}$

where $\lambda_{F}(p)$ (resp. $\lambda_{f}(p)$) are the normalized Hecke
eigenvalues of $F$ (resp., $f$). The relation (1) clearly shows the positivity
of the quantities $\lambda_{F}(p)$ (since we know that $|\lambda_{f}(p)|\leq
2$). A simple argument extends this positivity result to $\lambda_{F}(n)$ for
all $n$. Therefore, the topic of _signs of Hecke eigenvalues_ of Hecke
eigenforms in $S_{k}({\rm Sp}_{4}({\mathbb{Z}}))$ is interesting only for
forms that lie in the orthogonal complement111It is easy to see that if a
Hecke eigenform does not lie in $S_{k}^{*}({\rm Sp}_{4}({\mathbb{Z}}))$, then
it must be contained in the orthogonal complement of $S_{k}^{*}({\rm
Sp}_{4}({\mathbb{Z}}))$. of $S_{k}^{*}({\rm Sp}_{4}({\mathbb{Z}}))$.

For general $N$, we define a natural generalization of the Maass subspace
which we call the _generalized Maass subspace_ and denote by
$S_{k}^{*}({\Gamma}_{0}^{(2)}(N))$. The space
$S_{k}^{*}({\Gamma}_{0}^{(2)}(N))$ is defined to be the span of certain Hecke
eigenforms $F$ that we say are of _classical Saito-Kurokawa type_. These
eigenforms of classical Saito-Kurokawa type are defined by us in the language
of Arthur packets (see Section 2.3) and turn out to have the property that
their Hecke eigenvalues at primes $p\nmid N$ are all positive. We remark that
there is also a separate explicit construction of Saito-Kurokawa lifts for
${\Gamma}_{0}^{(2)}(N)$ for general $N$ via the route of Jacobi forms, due to
Ibukiyama [Ibu12] (see also Agarwal–Brown [AB15]). To the best of our
knowledge it is not known whether the Saito-Kurokawa lifts of trivial
character constructed by Ibukiyama are sufficient to fully capture222It can be
shown that if $N$ is _squarefree_ then the level $N$ newforms that are of
classical Saito-Kurokawa type in our sense are exactly the same as the
Saito–Kurokawa lifts of newforms of level $N$ constructed via Jacobi forms in
works of Ibukiyama and Agarwal–Brown; so the main question is what happens for
non-squarefree levels. our generalized Maass space
$S_{k}^{*}({\Gamma}_{0}^{(2)}(N))$ as defined in Section 2.3.

As mentioned above, for eigenforms of classical Saito-Kurokawa type the Hecke
eigenvalues at primes not dividing the level are all positive. Consider an
eigenform that is _not_ of classical Saito-Kurokawa type. Our first result
provides a bound in the $k$-aspect for the smallest prime at which the Hecke
eigenvalue is negative.

######  Theorem 1.1.1.

Let $k$ and $N$ be positive integers. Let $F\in S_{k}({\Gamma}_{0}^{(2)}(N))$
be a Hecke eigenform with Hecke eigenvalues $\eta(m)$ for all positive
integers $m$ satisfying $\gcd(m,N)=1$. Assume that $F$ does not lie in
$S_{k}^{*}({\Gamma}_{0}^{(2)}(N))$. If $k=2$ and $F$ is not attached to a CAP
representation, then assume further that $F$ satisfies the Ramanujan
conjecture.333The Ramanujan conjecture is known for all weight $k\geq 3$.
Readers unfamiliar with the definition of a CAP representation or the
Ramanujan conjecture should refer to Sections 2.2 and 2.3 . Then, for any
$\epsilon>0$, there exists a prime $p$ with $\gcd(p,N)=1$ and

$p~{}\ll_{N,\epsilon}~{}k^{5+\epsilon}$

such that $\eta(p)<0$. The implicit constant depends polynomially444Our method
can be used to make this polynomial explicit by carefully going through the
proof of Lemma 2.4.1. on $N$.

We also prove that for eigenforms not of classical Saito-Kurokawa type, the
associated Hecke eigenvalues $\eta(p)$ have infinitely many sign changes as
$p$ traverses a typical arithmetic progression. To the best of our knowledge,
this is the first result of this kind for Siegel cusp forms of degree 2.

######  Theorem 1.1.2.

Let $F$ be as in Theorem 1.1.1. Then, for any positive integers $a,M$ with
$\gcd(aN,M)=1$, the sequence $(\eta(p))$ with $p$ varying over primes
congruent to $a{\,\rm mod\,}M$ changes sign infinitely often.

Let us discuss briefly previous results related to the above theorems. The
investigation of first negative Hecke eigenvalue of Siegel cusp forms was
initiated by Kohnen and Sengupta [KS07] (for the case $N=1$). Exploring
relations among Hecke eigenvalues at prime power indices they proved the
existence of $n\ll k^{2}\log^{20}k$ such that $\eta(n)<0$. Note that in their
result $n$ can be any integer, in contrast to our Theorem 1.1.1 where $n$ is a
prime. The study of signs of Hecke eigenvalues at all integers has a different
flavour (and is usually easier) from studying sign-changes over the primes. We
remark here that the Hecke algebra is generated not just by $T(p)$ but by
$T(p)$ and $T(p^{2})$, so going down from all integers to primes is non-
trivial. Assuming the Generalized Lindelöf hypothesis one can improve the
bound obtained by Kohnen and Sengupta to $n\ll_{\epsilon}k^{\epsilon}$ for any
$\epsilon>0$. The result of Kohnen and Sengupta was generalized by Brown
[Bro10] to the case $N>1$. Other results on sign changes of Hecke eigenvalues
include [Koh07, KS07, PS08, RSW16, DK18, GKP21]. The first work that showed
that the sequence $\eta(p)$ for $p$ _prime_ has infinitely many sign changes
is due to Das [Das13] (however, he only considered the case $N=1$ and did not
consider primes in arithmetic progressions or bounds for the first sign-
change).

For more general automorphic $L$-functions, Cho and Kim [CK21, Theorem 1.1]
(building upon work of Jin [Jin19]) proved a bound for the first negative
coefficients at primes of $L$-function attached to a self-dual automorphic
representation under the assumption of Ramanujan conjecture. As a direct
consequence of [CK21, Theorem 1.1], one can find a prime $p\ll k^{7+\epsilon}$
such that $\eta(p)<0$, where $\eta(p)$ is as in Theorem 1.1.1. However, our
result in Theorem 1.1.1 gives a stronger bound as we have more precise
information about the archimedean $L$-factor in our case.

###  1.2. Main Results on Fourier coefficients

For Siegel cusp forms of degree $2$, the question of sign changes for the full
sequence of Fourier coefficients $a(T)~{}(T\in{\mathcal{N}}_{2})$ has been
investigated in [Jes08, CGK15, GS17]. However there are not many previous
results for the more subtle question of sign changes when $T$ is restricted to
some sparse subsequence of ${\mathcal{N}}_{2}$. A matrix
$T\in{\mathcal{N}}_{2}$ is called fundamental if $-\text{det }(2T)$ is a
fundamental discriminant, and in this case, the Fourier coefficient $a(T)$
will be called a fundamental Fourier coefficient. For a fixed
$T\in{\mathcal{N}}_{2}$ and varying $m\in{\mathbb{N}}$, we call the sequence
$a(mT)$ the radial Fourier coefficients associated to $T$. Very recently,
Jääsaari, Lester and Saha [JLS21, Theorem A] proved the infinitude of sign
changes for the subsequence of fundamental Fourier coefficients $a(T)$.
Shankhadhar and Tiwari [ST21] restricted themselves to certain other types of
Fourier coefficients $a(T)$. In this paper, we investigate signs of the radial
Fourier coefficients $a(pT)$ for fixed fundamental $T$ and $p$ varying over
primes. The properties of these coefficients $a(pT)$ are closely connected to
those of the Hecke eigenvalues $\eta(p)$.

Our result below (Theorem 1.2.1) gives a new characterisation of the forms of
classical Saito–Kurokawa type using sign changes of Fourier coefficients. We
note that our result does not require $F$ to be an eigenform. We refer the
reader to [PSS17, Section 8] for some alternative characterisations of Saito-
Kurokawa lifts.

######  Theorem 1.2.1 (see Propositions 4.3.1, 4.4.1).

Let $k$ and $N$ be positive integers. If $k=2$, assume that the Ramanujan
conjecture holds for each non-zero element of $S_{2}(\Gamma^{(2)}_{0}(N))$
that is a Hecke eigenform at all good primes and not attached to a CAP
representation.

Let $F\in S_{k}(\Gamma^{(2)}_{0}(N))$ be a non-zero cusp form having real
Fourier coefficients $a(T)$ and let $T_{0}$ be a fundamental matrix such that
$a(T_{0})\neq 0$ and $\gcd(4\det(T_{0}),N)=1$. If $F$ lies in
$S_{k}^{*}({\Gamma}_{0}^{(2)}(N))$, then for all sufficiently large primes $p$
the Fourier coefficients $a(pT_{0})$ have the same sign. On the other hand, if
$F$ is orthogonal to $S_{k}^{*}({\Gamma}_{0}^{(2)}(N))$, then there exist
infinitely many primes $p$ such that $a(pT_{0})>0$ and infinitely many primes
$q$ such that $a(qT_{0})<0$.

We remark here that given $F\in S_{k}(\Gamma^{(2)}_{0}(N))$, the existence of
(infinitely many) fundamental $T_{0}$ with $a(T_{0})\neq 0$ and
$\gcd(4\det(T_{0}),N)=1$ is known for $N$ squarefree under mild conditions on
$F$ at primes dividing $N$; see Remark 4.1.3 below.

Our final theorem on Fourier coefficients is quite different in flavour from
the results described so far and concerns upper bounds on the sizes of the
Fourier coefficients. Given $F\in S_{k}({\rm Sp}_{4}({\mathbb{Z}})$, the best
unconditional currently known upper bound for $a(T)$ is due to Kohnen [Koh92]
and states that $|a(T)|\ll_{F,\epsilon}({\rm
det}~{}T)^{\frac{k}{2}-\frac{13}{36}+\epsilon}$. However, this result is
rather far from the expected truth. The famous Resnikoff–Saldaña conjecture
[RSn74, Conjecture IV] predicts that for $F\in S_{k}({\rm
Sp}_{4}({\mathbb{Z}}))$ orthogonal to forms of classical Saito-Kurokawa type,
the Fourier coefficients satisfy the bound $|a(T)|\ll_{F,\epsilon}({\rm
det}~{}T)^{\frac{k}{2}-\frac{3}{4}+\epsilon}$ for any $\epsilon>0$. This
conjecture is extremely deep and difficult, and appears to be out of reach at
present. The article [DK13] proved bounds towards this for radial Fourier
coefficients while in [JLS21], the authors considered $F\in
S_{k}(\Gamma^{(2)}_{0}(N))$ with $N$ squarefree, and assuming the Generalized
Riemann Hypothesis proved the bound
$|a(T)|\ll_{F,\epsilon}\frac{\det(T)^{\frac{k}{2}-\frac{1}{2}}}{(\log|\det(T)|)^{\frac{1}{8}-\epsilon}}$
for fundamental matrices $T$.

For $F$ lying in the generalized Maass subspace
$S_{k}^{*}({\Gamma}_{0}^{(2)}(N))$, the situation is slightly different. In
this case one expects that the Resnikoff–Saldaña bound
$|a(T)|\ll_{F,\epsilon}({\rm det}~{}T)^{\frac{k}{2}-\frac{3}{4}+\epsilon}$
still holds for the fundamental Fourier coefficients $a(T)$ of $F$ (this can
be shown to follow from the Generalized Lindelöf hypothesis; see Proposition
5.1.5); however, it is known that this bound _fails_ to hold for general (non-
fundamental) coefficients. Indeed, Kohnen [Koh04] (see also Böcherer–Raghavan
[BR88]) observed that for a Saito-Kurokawa lift of full level, infinitely many
of the Fourier coefficients $a(LT_{0})$ $(L\geq 1)$ do _not_ satisfy the
Resnikoff–Saldaña bound. This was refined by Das and Kohnen [DK13] who showed
that given a Saito-Kurokawa lift of full level, there exists a fundamental
$T\in{\mathcal{N}}_{2}$ such that $|a(pT)|\gg_{F,T}p^{k-1}$ for $p$ lying in a
set of primes of positive density. Therefore, the Resnikoff–Saldaña conjecture
needs to be modified for forms in the Maass subspace. For the case $N=1$, Das
and Kohnen [DK13, section 4] conjectured that the bound

(2) $|a(T)|~{}\ll_{F,\epsilon}~{}\big{(}\det
T\big{)}^{\frac{k-1}{2}+\epsilon}$

holds for all Fourier coefficients $a(T)$ of a form lying in the Maass
subspace. From the examples above, it is clear that the exponent
$\frac{k-1}{2}$ in (2) is best possible. Our next result shows that (2) holds
for a form lying in the generalized Maass subspace of level $N$ under a mild
assumption on $\det(T)$ which is always satisfied when $N$ is squarefree.

######  Theorem 1.2.2.

Let $k$ and $N$ be positive integers. Let $F\in
S_{k}^{*}(\Gamma^{(2)}_{0}(N))$ be a non-zero cusp form having Fourier
coefficients $a(T)$. Then, for any $\epsilon>0$, the bound

$|a(T)|~{}\ll_{F,\epsilon}~{}\big{(}\det T\big{)}^{\frac{k-1}{2}+\epsilon}$

holds for all $T\in{\mathcal{N}}_{2}$ which have the property that
$\gcd(4\det(T),N)$ is squarefree.

In fact, our result is sharper than stated in Theorem 1.2.2, and gives a more
refined bound on the size of the Fourier coefficient; see Theorem 5.1.6. In
particular, Theorem 5.1.6 implies that for all primitive $T$ (i.e., there
being no common factor dividing the entries of $T$) we have
$|a(T)|\ll_{F,\epsilon}({\rm det}~{}T)^{\frac{k}{2}-\frac{3}{4}+\epsilon}$
under the Generalized Lindelöf hypothesis. This shows that the counterexamples
to the Resnikoff–Saldaña bound essentially arise from the matrices of the form
$LT_{0}$ as in the examples of Kohnen and Das–Kohnen noted above.

After a previous version of this manuscript was made available, we became
aware (thanks to Patrick Dynes) of a recent preprint by Ikeda and Katsurada
[IK22] where they prove a similar bound for the Fourier coefficients in the
overlapping context of an Ikeda lift of full level. Their methods are quite
different from ours and use the Gross–Keating invariant.

###  1.3. Methods

Theorems 1.1.1 and 1.1.2 rely on properties of the underlying $L$-functions.
More precisely, for Theorem 1.1.1, we build upon a method of Jin [Jin19,
Theorem 3.1] (also see Cho and Kim [CK21]) and for Theorem 1.1.2 we extend
techniques of Kohnen–Lau–Wu [KLW13] by adding a character twist. For both
these results, it is useful to separate into cases according to the various
global Arthur packets that $F$ can correspond to (see Proposition 2.3.1). If
$F$ is attached to a CAP representation, then we are essentially reduced to
properties of Dirichlet characters, which gives us stronger versions of the
desired results. On the other hand, if $F$ is not attached to a CAP
representation, then we use the transfer to ${\rm GL}_{4}$ and properties of
the (twists of the) ${\rm GL}_{4}$ $L$-functions to obtain these theorems.

The proof of Theorem 1.2.1 relies on two main ingredients. The first is an
identity relating Fourier coefficients and Hecke eigenvalues (see Proposition
4.1.1). The second ingredient is a Selberg orthogonality style relation for
_linear combinations_ of Hecke eigenvalues (Proposition 4.2.1) which exhibits
a certain dichotomy due to the nature of Arthur packets on ${\rm GSp}_{4}$ and
may be of independent interest.

For Theorem 1.2.2 we use quite different methods from the above. In fact, we
give two different proofs of this theorem. The first proof uses
representation-theoretic methods. Precisely, we use the spherical $p$-adic
Bessel functions to relate the size of $a(T)$ to that of $a(T_{0})$ where
$T_{0}$ is a fundamental matrix. This reduction requires delicate estimates
for the sizes of relevant values of these $p$-adic Bessel functions. The
desired bound for $a(T_{0})$ is a consequence of the convexity bound for
certain $L$-functions. As noted earlier, this method gives us a sharper result
than we have stated above; see Theorem 5.1.6 and Proposition 5.1.7 for the
more refined bounds.

The second proof is only valid for squarefree $N$ and is easy to derive from
recent developments in the theory of Fourier–Jacobi coefficients of a Siegel
cusp form. This proof exploits bounds relating the Petersson norm of a Jacobi
form, the sizes of its Fourier coefficients, and the index of the Jacobi form.
We decided to include both proofs here as they use very different ideas and
the second proof highlights the importance (as well as the limitations) of the
theory of Jacobi forms in studying Siegel modular forms. See Section 5 for
further details.

Some key preliminary facts will be listed in Section 2. In sections 3 – 6, we
give proofs of Theorems 1.1.1 – 1.2.2.

###  1.4. Notations

Throughout the paper, $\epsilon>0$ will denote a sufficiently small real
number whose precise value may change in each appearance. The notation
$A\ll_{x,y,z}B$ or $A=O_{x,y,z}(B)$ will mean there exists a positive constant
$C$ depending at most on $x,y,z$ such that $|A|\leq C|B|$.

We say that an integer $d$ is a fundamental discriminant if
${\mathbb{Q}}(\sqrt{d})$ is a field of discriminant $d$. For a fundamental
discriminant $d$, we let $\chi_{d}$ be the associated quadratic Dirichlet
character. For positive integers $a$, $b$, we use the notation
$\gcd(a,b^{\infty})$ to denote the limit
$\lim_{N\rightarrow\infty}\gcd(a,b^{N})$, which is clearly equal to
$\gcd(a,b^{N})$ for all sufficiently large $N$. For a prime $p$, we let
$v_{p}(a)$ denote the largest non-negative integer $v$ such that $p^{v}$
divides $a$.

We use ${\mathbb{A}}$ to denote the ring of adeles over ${\mathbb{Q}}$. The
notation $L(s,\pi)$ for the $L$-function of an automorphic representation
$\pi$ will mean the finite part of the $L$-function (i.e., without the
archimedean factors) and the notation $\Lambda(s,\pi)$ will denote the
completed $L$-function including the archimedean factors. All $L$-functions
will be normalized so that the functional equation takes $s\mapsto 1-s$. Given
an $L$-function with an Euler factor decomposition
$L(s,\pi)=\prod_{p}L(s,\pi_{p})$, we use the notation
$L^{N}(s,\pi):=\prod_{p\nmid N}L(s,\pi_{p})$ for each integer $N$ to denote
the partial $L$-function away from primes dividing $N$.

By $1_{2}$ we mean the identity matrix of size $2$. We denote by $J$ the $4$
by $4$ matrix given by $J=\left(\begin{smallmatrix}0&1_{2}\\\ -1_{2}&0\\\
\end{smallmatrix}\right),$ and we define the algebraic groups ${\rm GSp}_{4}$
and ${\rm Sp}_{4}$ by

${\rm GSp}_{4}(R)=\\{g\in{\rm
GL}_{4}(R)\;|\;g^{t}Jg=\mu_{2}(g)J,\>\mu_{2}(g)\in R^{\times}\\},$ ${\rm
Sp}_{4}(R)=\\{g\in{\rm GSp}_{4}(R)\;|\;\mu_{2}(g)=1\\},$

for any commutative ring $R$. The Siegel upper-half space $\mathbb{H}_{2}$ of
degree 2 is defined by

$\mathbb{H}_{2}=\\{Z\in\mathrm{Mat}_{2\times 2}({\mathbb{C}})\;|\;Z=Z^{t},\
\mathrm{Im}(Z)\text{ is positive definite}\\}.$

The subgroup ${\rm GSp}_{4}({\mathbb{R}})^{+}$ of ${\rm
GSp}_{4}({\mathbb{R}})$ consists of the matrices $g$ such that $\mu_{2}(g)>0$.
For $g=\left(\begin{smallmatrix}A&B\\\ C&D\end{smallmatrix}\right)\in{\rm
GSp}_{4}({\mathbb{R}})^{+}$ and $Z\in\mathbb{H}_{2}$, we define $g\langle
Z\rangle=(AZ+B)(CZ+D)^{-1}$ and we denote $J(g,Z)=CZ+D$. For a function
$F:\mathbb{H}_{2}\rightarrow{\mathbb{C}}$, a matrix $g\in{\rm
GSp}_{4}({\mathbb{R}})^{+}$, and an integer $k$, we define the function
$F|_{k}g:\mathbb{H}_{2}\rightarrow{\mathbb{C}}$ by
$(F|_{k}g)(Z)=\mu(g)^{k}\det(J(g,Z))^{-k}F(g\langle Z\rangle)$.

###  1.5. Acknowledgments

We thank Ariel Weiss and Patrick Dynes for comments. At the first stages of
this work, BP was a JSPS postdoctoral fellow at Kyushu University, Japan and
was supported by JSPS KAKENHI Grant No. 19F19318. He would like to thank his
academic host Professor Masanobu Kaneko for the support. This work is
supported by the Engineering and Physical Sciences Research Council [grant
number EP/W001160/1].

##  2\. Preliminaries

###  2.1. Basics of Siegel cusp forms

Let $k$ and $N$ be positive integers. Let $S_{k}(\Gamma^{(2)}_{0}(N))$ denote
the space of Siegel cusp forms of weight $k$ and of degree $2$ for the group
${\Gamma}^{(2)}_{0}(N)$ defined by

(3) $\Gamma^{(2)}_{0}(N)={\rm
Sp}_{4}({\mathbb{Z}})\cap\left(\begin{smallmatrix}{\mathbb{Z}}&{\mathbb{Z}}&{\mathbb{Z}}&{\mathbb{Z}}\\\
{\mathbb{Z}}&{\mathbb{Z}}&{\mathbb{Z}}&{\mathbb{Z}}\\\
N{\mathbb{Z}}&N{\mathbb{Z}}&{\mathbb{Z}}&{\mathbb{Z}}\\\
N{\mathbb{Z}}&N{\mathbb{Z}}&{\mathbb{Z}}&{\mathbb{Z}}\\\
\end{smallmatrix}\right).$

Recall that the elements of $S_{k}(\Gamma^{(2)}_{0}(N))$ consist of
holomorphic functions $F$ on $\mathbb{H}_{2}$ which satisfy the relation

(4) $F|_{k}\gamma=F\quad\text{for all }\gamma\in\Gamma^{(2)}_{0}(N),$

and vanish at all the cusps. For a precise formulation of this cusp vanishing
condition, see [Fre91, I.4.6]; for definitions and basic properties of Siegel
modular forms we refer the reader to [AZ95]. Given $F_{1}$ and $F_{2}$ in
$S_{k}(\Gamma^{(2)}_{0}(N))$, we define their Petersson inner product by

(5) $\langle F_{1},F_{2}\rangle=\frac{1}{[{\rm
Sp}(4,{\mathbb{Z}}):\Gamma_{0}^{(2)}(N)]}\;\int\limits_{\Gamma_{0}^{(2)}(N)\backslash\mathbb{H}_{2}}F_{1}(Z)\overline{F_{2}(Z)}(\det
Y)^{k-3}\,dX\,dY.$

For any positive integer $m\in{\mathbb{N}}$ with $\gcd(m,N)=1$, the Hecke
operator $T(m)$ acting on the space $S_{k}(\Gamma^{(2)}_{0}(N))$ is defined by

$T(m)F~{}=~{}m^{k-3}\sum_{\gamma\in{\Gamma}_{0}^{(2)}(N)\backslash\mathcal{O}_{2,m}(N)}F|_{k}\gamma,$

where

$\mathcal{O}_{2,m}(N)~{}:=~{}\left\\{\gamma\in\left(\begin{smallmatrix}{\mathbb{Z}}&{\mathbb{Z}}&{\mathbb{Z}}&{\mathbb{Z}}\\\
{\mathbb{Z}}&{\mathbb{Z}}&{\mathbb{Z}}&{\mathbb{Z}}\\\
N{\mathbb{Z}}&N{\mathbb{Z}}&{\mathbb{Z}}&{\mathbb{Z}}\\\
N{\mathbb{Z}}&N{\mathbb{Z}}&{\mathbb{Z}}&{\mathbb{Z}}\\\
\end{smallmatrix}\right)~{}|~{}\gamma^{t}J\gamma=mJ\right\\}.\phantom{m}$

We say that $F\in S_{k}(\Gamma^{(2)}_{0}(N))$ is a _Hecke eigenform at all
good primes_ if there exist complex numbers $\eta(m)$ such that
$T(m)F=\eta(m)F$ for all $m\in{\mathbb{N}}$ satisfying $\gcd(m,N)=1$. Using
the self-adjointness of the Hecke operators (see, e.g., Proposition 1.8 of
Chapter 4 of [AZ95]) it follows that the Hecke eigenvalues $\eta(m)$
associated to a Hecke eigenform are in fact all _real_ numbers. As in the case
of elliptic modular forms, the Hecke eigenvalues $\eta(m)$ are multiplicative.
We also use the normalized Hecke eigenvalues which are defined as:

(6)
$\lambda(m)~{}:=~{}\frac{\eta(m)}{m^{k-3/2}}\phantom{m}\text{for}\phantom{m}m\in{\mathbb{N}}.$

###  2.2. Automorphic representations of ${\rm GSp}_{4}({\mathbb{A}})$ and
$L$-functions

Given $F\in S_{k}(\Gamma^{(2)}_{0}(N))$, we let $\pi_{F}$ denote the
representation of ${\rm GSp}_{4}({\mathbb{A}})$ generated by the adelization
(in the sense of [Sah15, §3]) of $F$. The representation $\pi_{F}$ is of
trivial central character and can be written as a direct sum of irreducible,
cuspidal, automorphic representations of ${\rm GSp}_{4}({\mathbb{A}})$ of
trivial central characters. We denote by $\Pi(F)$ the set of irreducible
automorphic representations of ${\rm GSp}_{4}({\mathbb{A}})$ that occur in the
above direct sum decomposition of $\pi_{F}$. Clearly all elements of $\Pi(F)$
are also of trivial central character. As a consequence of multiplicity one
for ${\rm GSp}_{4}$ due to Arthur [Art13], it follows that if $\pi\in\Pi(F)$
then $\pi$ occurs in $\pi_{F}$ with multiplicity one. We say that an element
$F$ of $S_{k}(\Gamma^{(2)}_{0}(N))$ gives rise to an irreducible
representation if $\pi_{F}$ is irreducible, or equivalently, if $\Pi(F)$ is a
singleton set.

It can be seen easily [Sah15, Prop. 3.12] that if $F$ gives rise to an
irreducible representation then $F$ is a Hecke eigenform at all good primes.
While the converse is not true (due to failure of strong multiplicity one), we
have the following lemma which clarifies the situation.

######  Lemma 2.2.1.

Let $F\in S_{k}(\Gamma^{(2)}_{0}(N))$. Then the following conditions are
equivalent:

  1. (1)

If $\pi_{1}$ and $\pi_{2}$ are any two elements of $\Pi(F)$ then
$\pi_{1,p}\simeq\pi_{2,p}$ for all primes $p\nmid N$.

  2. (2)

If $\pi_{1}$ and $\pi_{2}$ are any two elements of $\Pi(F)$ then
$\pi_{1,p}\simeq\pi_{2,p}$ for almost all primes $p\nmid N$.

  3. (3)

$F$ is a Hecke eigenform at all good primes, i.e., all primes not dividing
$N$.

  4. (4)

$F$ is a Hecke eigenform at almost all good primes, i.e., there exits a
multiple $N^{\prime}$ of $N$ such that $F$ is an eigenform for the Hecke
operators $T(m)$ for all $m\in{\mathbb{N}}$ satisfying $\gcd(m,N)=1$.

###### Proof.

The equivalence of (1) and (2) follows from Lemma 3.1.2 of [Sch05], the fact
that $\pi_{i,p}$ is spherical for all $p\nmid N$, and the fact that there is a
unique spherical constituent in each induced representation. The equivalence
of (1) and (3), as well as of (2) and (4), follows from Proposition 3.12 of
[Sah15]. ∎

Given $F\in S_{k}(\Gamma^{(2)}_{0}(N))$, and $\pi\in\Pi(F)$, we can take the
projection $\phi^{\prime}_{F}$ of the adelization $\phi_{F}$ of $F$ onto the
$\pi$-isotypic subspace of $\pi_{F}$, and then unadelize $\phi^{\prime}_{F}$
to obtain some $F^{\prime}\in S_{k}(\Gamma^{(2)}_{0}(N))$ that is a Hecke
eigenform at all good primes and whose adelization generates the space
$V_{\pi}$ of $\pi$. _Therefore, given an irreducible cuspidal automorphic
representation $\pi$ of ${\rm GSp}_{4}({\mathbb{A}})$, there exists $F\in
S_{k}(\Gamma^{(2)}_{0}(N))$ such that $\pi=\pi_{F}$ if and only if there
exists $F\in S_{k}(\Gamma^{(2)}_{0}(N))$ such that $\pi\in\Pi(F)$._ If $\pi$
has the above property, we say that $\pi$ arises from
$S_{k}(\Gamma^{(2)}_{0}(N))$.

We say that an irreducible cuspidal automorphic representation $\pi$ of ${\rm
GSp}_{4}({\mathbb{A}})$ is CAP if it is nearly equivalent to a constituent of
a global induced representation of a proper parabolic subgroup of ${\rm
GSp}_{4}({\mathbb{A}})$. If $F\in S_{k}(\Gamma^{(2)}_{0}(N))$ is a Hecke
eigenform at all good primes, we say that $F$ is CAP (or $F$ is attached to a
CAP representation) if some (equivalently, every) $\pi\in\Pi(F)$ is CAP.

Given an irreducible cuspidal automorphic representation $\pi$ of ${\rm
GSp}_{4}({\mathbb{A}})$ of trivial central character, we let

(7) $L(s,\pi)=\prod_{p<\infty}L(s,\pi_{p})$

denote the (finite part of the) degree $4$ (spinor) $L$-function associated to
$\pi$, where the local factors $L(s,\pi_{p})$ are defined using the local
Langlands correspondence [GT11] for ${\rm GSp}_{4}$. For each prime $p\nmid
N$, there are complex numbers (known as Satake parameters) $\alpha_{p,i}$,
$1\leq i\leq 4$ satisfying
$\alpha_{p,1}\alpha_{p,2}=\alpha_{p,3}\alpha_{p,4}=1$ such that
$L(s,\pi_{p})=\prod_{i=1}^{4}\left(1-\frac{\alpha_{p,i}}{p^{s}}\right)^{-1}$.
By the work of Arthur [Art13] and using the accidental isomorphism
$\mathrm{PGSp}_{4}\simeq\mathrm{SO}_{5}$, we know that
$L^{N}(s,\pi)=L^{N}(s,\Pi)$ where $\Pi$ is an isobaric automorphic
representation of ${\rm GL}_{4}({\mathbb{A}})$.

Given two irreducible cuspidal automorphic representation $\pi_{1}$ and
$\pi_{2}$ of ${\rm GSp}_{4}({\mathbb{A}})$ of trivial central characters, we
let

(8) $L(s,\pi_{1}\times\pi_{2})=\prod_{p<\infty}L(s,\pi_{1,p}\times\pi_{2,p})$

denote the associated degree $16$ Rankin-Selberg product of the spin
$L$-functions of $\pi_{1}$ and $\pi_{2}$, where each local factor is defined
using the local Langlands correspondence. For each prime $p\nmid N$, we have
$L(s,\pi_{1,p}\times\pi_{2,p})=\prod_{1\leq i,j\leq
4}\left(1-\alpha_{p,i}\beta_{p,j}p^{-s}\right)^{-1}$. As before, using work of
Arthur, we have that
$L^{N}(s,\pi_{1}\times\pi_{2})=L^{N}(s,\Pi_{1}\times\Pi_{2})$ where $\Pi_{1}$
and $\Pi_{2}$ are isobaric automorphic representations of ${\rm
GL}_{4}({\mathbb{A}})$. By the general theory of Rankin-Selberg $L$-functions
on ${\rm GL}_{n}$, the $L$-function $L(s,\pi_{1}\times\pi_{2})$ has analytic
continuation to ${\mathbb{C}}$ as a meromorphic function which is non-
vanishing on $\mathrm{Re}(s)=1$.

Let $F\in S_{k}(\Gamma^{(2)}_{0}(N))$ be a Hecke eigenform at all good primes
and let $\pi\in\Pi(F)$. Then we have the following relation between the degree
$4$ $L$-function of $\pi$ and the normalized Hecke eigenvalues of $F$:

(9) $L^{N}(s,\pi)~{}=~{}\zeta^{N}(2s+1)\sum_{\begin{subarray}{c}m=1\\\
(m,N)=1\end{subarray}}^{\infty}\frac{\lambda(m)}{m^{s}}~{}=~{}\prod_{p\nmid
N}\prod_{i=1}^{4}\left(1-\frac{\alpha_{p,i}}{p^{s}}\right)^{-1}.$

###  2.3. Arthur packets

Cuspidal automorphic representation of ${\rm GSp}_{4}$ with trivial central
character can be naturally viewed as representations of ${\rm
PGSp}_{4}\simeq{\rm SO}_{5}$. Arthur [Art13] has given a classification of the
discrete automorphic spectrum of ${\rm SO}_{5}$ into _Arthur packets_ in terms
of automorphic representations of general linear groups. In the following
proposition, we state this classification for the $\pi$ which arise from
$S_{k}(\Gamma^{(2)}_{0}(N))$ and write some key properties for each packet.

######  Proposition 2.3.1.

Suppose that $F\in S_{k}(\Gamma^{(2)}_{0}(N))$ is a Hecke eigenform at all
good primes. Let the normalized Hecke eigenvalues $\lambda(m)$ of $F$ be as
defined in (6). Then exactly one of the following cases must occur.

  1. (1)

$F$ is of _general_ type: This case can only occur if $k\geq 2$. There exists
an irreducible _cuspidal_ automorphic representation $\Pi$ of ${\rm
GL}_{4}({\mathbb{A}})$ with trivial central character such that each
$\pi\in\Pi(F)$ has a strong functorial lifting to $\Pi$. In this case,
$L(s,\pi)=L(s,\Pi)$ is entire and $L(s,\pi\times\pi)=L(s,\Pi\times\Pi)$ has a
simple pole at $s=1$. The representation $\Pi$ is unramified at the finite
primes not dividing $N$.

  2. (2)

$F$ is of _Yoshida_ type: This case can only occur if $N>1$ and $k\geq 2$.
There exists an automorphic representation $\pi_{1}$ of ${\rm
GL}_{2}({\mathbb{A}})$ attached to a cuspidal newform $f_{1}$ of weight $2$
for $\Gamma_{0}(N_{1})$ and an automorphic representation $\pi_{2}$ of ${\rm
GL}_{2}({\mathbb{A}})$ attached to a cuspidal newform $f_{2}$ of weight $2k-2$
for $\Gamma_{0}(N_{2})$, with $\pi_{1}$ and $\pi_{2}$ non-isomorphic, such
that for each $\pi\in\Pi(F)$, we have $L(s,\pi)=L(s,\pi_{1})L(s,\pi_{2})$.
Thus, the functorial lift of the representation $\pi$ to ${\rm
GL}_{4}({\mathbb{A}})$ is the isobaric sum of two distinct, unitary, cuspidal
automorphic representations of ${\rm GL}_{2}({\mathbb{A}})$ with trivial
central characters and $L(s,\pi)$ is entire. Any prime $p$ dividing
$N_{1}N_{2}$ must divide $N$. For each prime $p$ coprime to $N$ we have
$\lambda(p)=\lambda_{1}(p)+\lambda_{2}(p)$ where $\lambda_{i}(p)$ is the
normalized Hecke eigenvalue of $f_{i}$ at the prime $p$. We have the
factorization

$L(s,\pi\times\pi)=L(s,\pi_{1}\times\pi_{2})^{2}\zeta(s)^{2}L(s,{\rm
sym}^{2}\pi_{1})L(s,{\rm sym}^{2}\pi_{2})$

and hence $L(s,\pi\times\pi)$ has a pole of order $2$ at $s=1$.

  3. (3)

$F$ is of _Saito–Kurokawa_ type: This case can only occur if $k\geq 2$;
moreover if $N$ is squarefree, then $k$ must be even. There exists a
representation $\pi_{0}$ of ${\rm GL}_{2}({\mathbb{A}})$ of trivial central
character attached to a cuspidal holomorphic newform of weight $2k-2$ and a
primitive Dirichlet character $\chi_{0}$ satisfying $\chi_{0}^{2}=1$ such that

$L^{N}(s,\pi)=L^{N}(s,\pi_{0})L^{N}(s+1/2,\chi_{0})L^{N}(s-1/2,\chi_{0})$

for each representation $\pi\in\Pi(F)$. In particular, each such $\pi$ is CAP
with respect to the Siegel parabolic. The representation $\pi_{0}$ and the
character $\chi_{0}$ are unramified at the finite primes not dividing $N$. For
each prime $p$ coprime to $N$ we have
$\lambda(p)=\lambda_{0}(p)+p^{1/2}\chi_{0}(p)+p^{-1/2}\chi_{0}(p)$ where
$\lambda_{0}(p)$ is the normalized Hecke eigenvalue of $\pi_{0}$ at the prime
$p$.

If $F$ is of Saito–Kurokawa type with $\chi_{0}$ trivial (which is _always_
the case if $N$ is squarefree), we will say that $F$ is of _classical Saito-
Kurokawa_ type.

  4. (4)

$F$ is of _Soudry_ type: This case can only occur if $k\in\\{1,2\\}$ and
$N>1$. There exists a unitary cuspidal automorphic representation $\pi_{0}$ of
${\rm GL}_{2}({\mathbb{A}})$ attached to a CM cuspidal holomorphic newform
$f_{0}$ of weight 1, level $N_{0}$ and character $\xi_{0}$ such that

$L^{N}(s,\pi)=L^{N}(s+1/2,\pi_{0})L^{N}(s-1/2,\pi_{0})$

for each $\pi\in\Pi(F)$. Therefore each representation $\pi\in\Pi(F)$ is CAP
with respect to the Klingen parabolic. The character $\xi_{0}$ is a non-
trivial quadratic Dirichlet character of conductor dividing $N_{0}$, and any
prime $p$ dividing $N_{0}$ must divide $N$. For each prime $p$ coprime to $N$
we have $\lambda(p)=p^{1/2}\lambda_{0}(p)+p^{-1/2}\lambda_{0}(p)$ where
$\lambda_{0}(p)$ is the normalized Hecke eigenvalue of $\pi_{0}$ at the prime
$p$.

  5. (5)

$F$ is of _Howe–Piatetski-Shapiro_ type: This case can only occur if
$k\in\\{1,2\\}$ and $N>1$. Each representation $\pi\in\Pi(F)$ is CAP with
respect to the Borel parabolic. There exist distinct quadratic Dirichlet
characters $\chi_{1}$ and $\chi_{2}$ such that

$L^{N}(s,\pi)=L^{N}(s+1/2,\chi_{1})L^{N}(s-1/2,\chi_{1})L^{N}(s+1/2,\chi_{2})L^{N}(s-1/2,\chi_{2}).$

The representation $\chi_{1}$ and $\chi_{2}$ are unramified at the finite
primes not dividing $N$. For each prime $p$ coprime to $N$ we have
$\lambda(p)=p^{1/2}\chi_{1}(p)+p^{-1/2}\chi_{1}(p)+p^{1/2}\chi_{2}(p)+p^{-1/2}\chi_{2}(p).$

###### Proof.

This follows from the work of Arthur [Art13]; see also the papers of Schmidt
[Sch18, Sch20]. For the convenience of the reader, we explain some of the key
points below.

Let $\pi\in\Pi(F)$. Then, as explained in Section 1.1 of [Sch18], the global
(Arthur) parameter $\psi$ of $\pi$ in the sense of [Art13] is equal to a
formal expression of the form $\sum_{i=1}^{r}\mu_{i}\boxplus\nu(i)$ where
$\mu_{i}$ is a self-dual, unitary, cuspidal, automorphic representation of
${\rm GL}_{m_{i}}({\mathbb{A}})$, $\nu(i)$ is the irreducible representation
of ${\rm SL}_{2}({\mathbb{C}})$ of dimension $n_{i}$, and
$\sum_{i=1}^{r}m_{i}n_{i}=4$. Using strong multiplicity 1 for ${\rm
GL}_{n}({\mathbb{A}})$, it is easy to see that the global parameter does not
depend on which $\pi\in\Pi(F)$ we choose; in particular the global parameter
$\psi$ depends only on $F$. The local component $\pi_{v}$ at each place $v$
belongs to the local packet associated to the corresponding local parameter
$\psi_{v}$. In particular, this means that if $p\nmid N$, then each
$\mu_{i,p}$ is unramified since $\pi_{p}$ is spherical.

The possibilities for the global parameter can be divided into the following
types where we use the notation in [Sch18]. We enumerate them in the same
order as in the statement of this proposition.

  1. (1)

$\psi=\mu\boxtimes 1$, where $\mu$ is a self-dual, symplectic, unitary,
cuspidal, automorphic representation of ${\rm GL}_{4}({\mathbb{A}})$. The
assertions follow by taking $\Pi=\mu$. The fact that $\pi$ has a strong
lifting to $\Pi$ is just the equivalence of local Arthur parameters and local
(Langlands) $L$-parameters in this setting; see the remarks before Theorem 1.1
of [Sch18]. The other assertions follow from standard properties of
$L$-functions on ${\rm GL}_{n}({\mathbb{A}})$.

  2. (2)

$\psi=\mu_{1}\boxtimes 1\boxplus\mu_{2}\boxtimes 1$ where $\mu_{1}$ and
$\mu_{2}$ are distinct, unitary, cuspidal, automorphic representation of ${\rm
GL}_{2}({\mathbb{A}})$ of trivial central characters. We put
$\pi_{i}=\mu_{i}$. From the archimedean local parameters, we obtain the
existence of newforms of weights 2 and $2k-2$ attached to $\pi_{1}$ and
$\pi_{2}$ as desired. Again, the equivalence of local Arthur parameters and
local (Langlands) $L$-parameters in this setting gives the desired expressions
for $L(s,\pi)$ and $L(s,\pi\times\pi)$. The formula for the Hecke eigenvalue
is immediate from this. Clearly $2k-2\geq 2$ and so $k\geq 2$.

  3. (3)

$\psi=(\mu\boxtimes 1)\boxplus(\sigma\boxtimes\nu(2))$ where $\mu$ is a
unitary, cuspidal, automorphic representation of ${\rm GL}_{2}({\mathbb{A}})$
of trivial central character and $\sigma$ is a quadratic Dirichlet character.
We take $\pi_{0}=\mu$ and $\chi_{0}=\sigma$. The fact that $\pi_{0}$
corresponds to a holomorphic form of weight $2k-2$, $k\geq 2$ follows
immediately by looking at the archimedean local parameters (see the last row
of Table 2 of [Sch20]). The expression for the $L$-functions (and the Hecke
eigenvalues) is a consequence of the corresponding identity relating the local
$L$-parameters at all primes not dividing $N$ (see the first row of Table 2 of
[Sch20]).

The criterion for $\pi$ to occur in the discrete spectrum is expressed by the
sign condition
$\prod_{v}\epsilon(\pi_{v})=\epsilon(1/2,\pi_{0}\times\chi_{0})$ coming from
Arthur’s multiplicity formula (see the condition above Lemma 1.2 of [Sch20]).
Above, $\epsilon_{v}(\pi_{v})\in\\{\pm 1\\}$ is a certain sign occurring in
Arthur’s work. Now assume that $N$ is squarefree. In this case $\chi_{0}$ must
be unramified everywhere by a well-known result of Borel [Bor76] and hence
must be trivial. This implies that in this case
$\prod_{v}\epsilon(\pi_{v})=\epsilon(1/2,\pi_{0}).$ We claim that $k$ must be
even in this case. To show this, let $N_{0}$ be the conductor of $\pi_{0}$.
Then by part (2) of Lemma 5.1.2 below, $N_{0}$ divides $N$ and for each
$p|N_{0}$ one of the following possibilities must hold:

     * •

$\pi_{p}\simeq\tau(T,\nu^{-1/2})$ is of type VIb and
$\pi_{0,p}=\mathrm{St}_{{\rm GL}(2)}$,

     * •

$\pi_{p}\simeq L((\nu^{1/2}\xi_{p}\mathrm{St}_{{\rm GL}(2)},\nu^{-1/2})$ is of
type Vb and $\pi_{0,p}=\xi_{p}\mathrm{St}_{{\rm GL}(2)}$.

Using the formulas for $\epsilon(\pi_{v})$ in Table 2 of [Sch20] we have for
all $p|N_{0}$ that $\epsilon(\pi_{p})=\epsilon(1/2,\pi_{0,p})$ (more
precisely, they are each equal to -1 in the first possiblity and +1 in the
second possibility). If $p$ is a finite prime not dividing $N_{0}$, we have
$\epsilon(\pi_{p})=\epsilon(1/2,\pi_{0,p})=1.$ It follows that
$\epsilon(\pi_{\infty})=\epsilon(1/2,\pi_{0,\infty})$. Since $\pi$ is
holomorphic of weight $k$, we have $\epsilon(\pi_{\infty})=-1$ by Table 2 of
[Sch20]; on the other hand since $\pi_{0}$ corresponds to a holomorphic form
of weight $2k-2$ we have $\epsilon(1/2,\pi_{0,\infty})=(-1)^{k-1}$.
Consequently, $(-1)^{k-1}=-1$ and so $k$ must be even whenever $N$ is
squarefree.

  4. (4)

$\psi=\mu\boxtimes\nu(2)$ where $\mu$ is a unitary, cuspidal, automorphic
representation of ${\rm GL}_{2}({\mathbb{A}})$ of non-trivial quadratic
central character satisfying $\mu=\mathrm{AI}_{E/{\mathbb{Q}}}(\theta)$ where
$E$ is the quadratic field attached to the central character $\xi_{0}$ of
$\mu$ and $\theta$ is a Hecke character of $E$. From the computation in Table
3 of [Sch20], we see that $k$ equals 1 or 2 and that $\mu$ is attached to a CM
cuspidal holomorphic newform of weight 1. Since the conductor $N_{0}$ of a CM
newforms is never equal to 1, and since each prime dividing $N_{0}$ must
divide $N$, it follows that $N>1$. We take $\pi_{0}=\mu$ and the expression
for the $L$-functions is a consequence of the corresponding identity of local
$L$-parameters at all primes not dividing $N$ (see Table 3 of [Sch20]).

  5. (5)

$\psi=(\chi_{1}\boxtimes\nu(2))\boxplus(\chi_{1}\boxtimes\nu(2))$ where
$\chi_{1}$ and $\chi_{2}$ are distinct quadratic Dirichlet characters. From
the computation in Table 1 of [Sch20], we see that $k$ equals 1 or 2 . The
expression for the $L$-functions (and as a consequence, the Hecke eigenvalues)
follows from the corresponding identity of local $L$-parameters at all primes
not dividing $N$ (see Table 1 of [Sch20]). Since at least one of the
$\chi_{i}$ is non-trivial, and since each prime dividing the conductor of
$\chi_{1}$ or $\chi_{2}$ must divide $N$, it follows that $N>1$.

Finally, we note that a result of Weissauer [Wei92] asserts that if $k=1$ then
$F$ must be attached to a CAP representation; in particular, it cannot be of
Type 1. ∎

If $F\in S_{k}(\Gamma^{(2)}_{0}(N))$ is a Hecke eigenform at good primes that
is either of Saito–Kurokawa type, or of Soudry type, or of Howe–Piatetski-
Shapiro type in the sense defined above then $\pi\in\Pi(F)$ is CAP and so is
non-tempered everywhere.

On the other hand if $F\in S_{k}(\Gamma^{(2)}_{0}(N))$ is a Hecke eigenform at
good primes that is of general type or Yoshida type in the sense defined
above, then one expects that the following conjecture is true.

######  Conjecture 2.3.2 (Ramanujan conjecture, currently known for $k\geq 3$
but open for $k=2$).

If $F\in S_{k}(\Gamma^{(2)}_{0}(N))$ is a Hecke eigenform at good primes that
is non-CAP (i.e., of general type or Yoshida type), then for $\gcd(p,N)=1$ we
have $|\alpha_{p,i}|~{}=~{}1$ for $1\leq i\leq 4$ where $\alpha_{p,i}$ are the
Satake parameters defined in (9).

A famous result of Weissauer [Wei09] implies that $F$ satisfies the Ramanujan
conjecture whenever $k\geq 3$. More precisely, for such $F$, each
$\pi\in\Pi(F)$ is _tempered_ at all good primes, i.e., for $\gcd(p,N)=1$ we
have $|\alpha_{p,i}|~{}=~{}1$ for $1\leq i\leq 4$ and hence for $\gcd(m,N)=1$,
$|\lambda(m)|\leq d_{5}(m)$, where $d_{5}(m)$ is the number of ways of writing
$m$ as product of $5$ positive integers. If $k=2$, and $F$ is of Yoshida type,
then we also know the Ramanujan conjecture for $F$ from our knowledge of the
Ramanujan conjecture for classical holomorphic cusp forms on the upper-half
plane.

However, if $k=2$ and $F\in S_{2}(\Gamma^{(2)}_{0}(N))$ is a Hecke eigenform
at good primes that is of general type, then the Ramanujan conjecture remains
open. For several of our results in this paper, we will need to assume the
Ramanujan conjecture in this outstanding case. Note that if $k=1$, then all
forms in $S_{1}(\Gamma^{(2)}_{0}(N))$ are CAP.

######  Remark 2.3.3.

If $F\in S_{2}(\Gamma^{(2)}_{0}(N))$ is a Hecke eigenform at good primes that
is of general type, then we know the bound $|\alpha_{p,i}|\leq p^{9/22}$ for
all primes $p$ with $\gcd(p,N)=1$, which follows from the corresponding bound
for ${\rm GL}_{4}$ representations due to Kim–Sarnak [Kim03].

From the self-adjointness of the Hecke operators, it follows that if $F_{1}$
and $F_{2}$ correspond to different cases (types) in the sense of Proposition
2.3.1, then $F_{1}$ and $F_{2}$ are orthogonal with respect to the Petersson
norm. We let $S_{k}(\Gamma^{(2)}_{0}(N))^{\rm T}$ equal the vector space
generated by the set of Hecke eigenforms at good primes which are of general
type or Yoshida type. We let $S_{k}(\Gamma^{(2)}_{0}(N))^{\rm CAP}$ equal the
vector space generated by the set of Hecke eigenforms at good primes $F$ which
are non-tempered, i.e., are either of Saito-Kurokawa type or of Soudry type or
of Howe– Piatetski-Shapiro type. So the space $S_{k}(\Gamma^{(2)}_{0}(N))$ has
a natural decomposition into orthogonal subspaces

$S_{k}(\Gamma^{(2)}_{0}(N))=S_{k}(\Gamma^{(2)}_{0}(N))^{\rm T}\oplus
S_{k}(\Gamma^{(2)}_{0}(N))^{\rm CAP}.$

We let $S_{k}^{*}(\Gamma^{(2)}_{0}(N))\subset S_{k}(\Gamma^{(2)}_{0}(N))^{\rm
CAP}$ be the subspace generated by the set of Hecke eigenforms at good primes
which are of classical Saito-Kurokawa type. We will occasionally refer to the
space $S_{k}^{*}(\Gamma^{(2)}_{0}(N))$ as the _generalized Maass subspace_.
From Proposition 2.3.1 it is clear that if $k>2$ and $N$ is squarefree then
$S_{k}^{*}(\Gamma^{(2)}_{0}(N))=S_{k}(\Gamma^{(2)}_{0}(N))^{\rm CAP}$.

###  2.4. Analytic conductors and convexity bounds

Given a prime $p$ and an irreducible admissible representation $\pi_{p}$ of
${\rm GSp}_{4}({\mathbb{Q}}_{p})$, we let $\epsilon(s,\pi_{p})$ denote the
epsilon factor of $\pi_{p}$ (with respect to a fixed additive character
$\psi_{p}$ of level 0), where the epsilon factor is defined using the local
Langlands correspondence (in particular, we have
$\epsilon(s,\pi_{p})=\epsilon(s,\Pi_{p})$ where $\Pi_{p}$ is the
representation on ${\rm GL}_{4}({\mathbb{Q}}_{p})$ that is obtained by local
Langlands transfer from $\pi_{p}$). We let
$\epsilon(s,\pi_{p}\times\pi_{p})=\epsilon(s,\Pi_{p}\times\Pi_{p})$ denote the
epsilon factor of the Rankin-Selberg $L$-function with respect to the same
fixed unramified additive character $\psi_{p}$. We define the conductor
exponents $a(\pi_{p})$ and $a(\pi_{p}\times\pi_{p})$ via

$\epsilon(s,\pi_{p})=\epsilon(1/2,\pi_{p})p^{-a(\pi_{p})(s-1/2)},$
$\epsilon(s,\pi_{p}\times\pi_{p})=\epsilon(1/2,\pi_{p}\times\pi_{p})p^{-a(\pi_{p}\times\pi_{p})(s-1/2)}.$

Since the conductor exponents above coincide with the corresponding conductor
exponents arising from representations of ${\rm GL}_{4}({\mathbb{Q}}_{p})$,
the results of [BH17] and [Cor19] imply that there exist absolute positive
constants $c_{1}$, $c_{2}$ and $d$ such that555With some bookkeeping, one can
show that it suffices to take $c_{2}=7$, $c_{1}=2$ and $d=64$. for any
irreducible admissible representation $\pi_{p}$ of ${\rm
GSp}_{4}({\mathbb{Q}}_{p})$ of trivial central character, we have the bound

(10) $c_{1}a(\pi_{p})-d\leq a(\pi_{p}\times\pi_{p})\leq c_{2}a(\pi_{p})$

Given an irreducible cuspidal automorphic representation
$\pi\simeq\otimes_{v}\pi_{v}$ of ${\rm GSp}_{4}({\mathbb{A}})$, we define its
finite conductor $q(\pi)$ by

(11) $q(\pi)=\prod_{p<\infty}p^{a(\pi_{p})}.$

We define the finite conductor of $\pi\times\pi$ by

(12) $q(\pi\times\pi)=\prod_{p<\infty}p^{a(\pi_{p}\times\pi_{p})}.$

The next lemma shows that the conductor is always bounded by a power of the
level. While this result is probably known to experts, we were unable to find
a suitable reference.

######  Lemma 2.4.1.

There exist absolute constants $C$ and $D$ such that given a positive integer
$N$ and an automorphic representation $\pi$ of ${\rm GSp}_{4}({\mathbb{A}})$
arising from $S_{k}(\Gamma^{(2)}_{0}(N))$, we have $q(\pi)\ll N^{C}$ and
$q(\pi\times\pi)\ll N^{D}.$

###### Proof.

Let $\pi$ be as in the Lemma. Then $\pi_{p}$ is unramified for all $p\nmid N$
and hence $a(\pi_{p})=0$ for such $p$. Consider a prime $p$ dividing $N$. Then
$\pi_{p}$ has a vector fixed by

(13) $K_{0,p}(N):={\rm
GSp}_{4}({\mathbb{Z}}_{p})\cap\left(\begin{smallmatrix}{\mathbb{Z}}_{p}&{\mathbb{Z}}_{p}&{\mathbb{Z}}_{p}&{\mathbb{Z}}_{p}\\\
{\mathbb{Z}}_{p}&{\mathbb{Z}}_{p}&{\mathbb{Z}}_{p}&{\mathbb{Z}}_{p}\\\
N{\mathbb{Z}}_{p}&N{\mathbb{Z}}_{p}&{\mathbb{Z}}_{p}&{\mathbb{Z}}_{p}\\\
N{\mathbb{Z}}_{p}&N{\mathbb{Z}}_{p}&{\mathbb{Z}}_{p}&{\mathbb{Z}}_{p}\\\
\end{smallmatrix}\right).$

Let $K_{p}(N)\subseteq K_{0,p}(N)$ be the principal congruence subgroup
defined by $K_{p}(N)=\\{g\in{\rm GSp}_{4}({\mathbb{Z}}_{p}):g\equiv
1_{4}\pmod{N}\\}$. It is clear that the lemma follows from (10) - (12) and the
following purely local statement: _There exists an absolute constant $C$ such
that for a prime $p$, a positive integer $n_{p}$ and an irreducible admissible
representation $\pi_{p}$ of ${\rm GSp}_{4}({\mathbb{Q}}_{p})$ that contains a
non-zero $K_{p}(p^{n_{p}})$-fixed vector, we have $a(\pi_{p})\leq Cn_{p}$._

We now give a proof of the local statement. By the well-known classification
theorem of irreducible admissible representations, we have the following two
cases

  * •

$\pi_{p}$ is obtained as a subquotient of a parabolically induced
representation from one of the parabolic subgroups of ${\rm GSp}_{4}$,

  * •

$\pi_{p}$ is a supercuspidal representation.

We consider the first case. In this case, the induced representation is
induced from a collection of characters of ${\rm GL}_{1}({\mathbb{Q}}_{p})$
and/or admissible representations of ${\rm GL}_{2}({\mathbb{Q}}_{p})$,
depending on the parabolic subgroup (see Chapter 2 of [RS07] for more
details). Since the induced representation has a vector fixed by
$K_{p}(p^{n_{p}})$, it follows easily using the Iwasawa decomposition that any
character occurring in the inducing data must have conductor exponent at most
$n_{p}$ and any representation of ${\rm GL}_{2}({\mathbb{Q}}_{p})$ occurring
in the inducing data must have a vector fixed by $\\{g\in{\rm
GL}_{2}({\mathbb{Z}}_{p}):g\equiv 1_{2}\pmod{p^{n_{p}}}\\}$ and hence must
have conductor exponent at most $2n_{p}$. Now the local statement follows
immediately from the direct expression of $a(\pi_{p})$ as a sum of the above
conductor-exponents (Table A.9 of [RS07]).

We now move on to the second case. In this case, using the fact that $\pi_{p}$
contains a non-zero $K_{p}(p^{n_{p}})$-fixed vector, it is immediate that the
formal degree ${\rm deg}(\pi_{p})$ [GI14, section 13] of $\pi_{p}$ satisfies
${\rm deg}(\pi_{p})\leq p^{Dn_{p}}$ for some absolute positive constant $D$.
On the other hand, the formal degree conjecture (which is proved for ${\rm
GSp}_{4}$ by the main result of [GI14] implies that ${\rm deg}(\pi_{p})\geq
p^{Ea(\pi_{p}\times\pi_{p})}$ for some absolute positive constant $E$. This
combined with (10) concludes the proof of the local statement in this case. ∎

Suppose $F\in S_{k}(\Gamma^{(2)}_{0}(N))^{\rm T}$ is a Hecke eigenform at good
primes and $\pi\in\Pi(F)$. We define the completed spinor $L$-function (see
Table 3 of [Sch17])

(14)
$\Lambda(s,\pi):=(2\pi)^{-2s-k+1}{\Gamma}(s+k-3/2){\Gamma}(s+1/2)L(s,\pi)$

which coincides with the completed $L$-function of an automorphic
representation of ${\rm GL}_{4}({\mathbb{A}})$ by the previous subsection and
can therefore be meromorphically extended to the whole complex plane and
satisfies the functional equation

$\Lambda(s,\pi)~{}=~{}\epsilon(s,\pi)\Lambda(1-s,\pi).$

Using (14), we see that the analytic conductor $\mathfrak{q}(s,\pi)$ of
$L(s,\pi)$ (for definition see [IK04, p.95]) is

$\mathfrak{q}(s,\pi)\ll q(\pi)k^{2}(|s|+3)^{4}\ll_{N}k^{2}(|s|+3)^{4}.$

Therefore the convexity bound for $L(s,\pi)$ in the weight/$t$-aspect is (see
[IK04, p.101])

(15)
$|L(1/2+\epsilon+it,\pi)|~{}\ll_{N,\epsilon}~{}k^{\frac{1}{2}+\epsilon}(3+|t|)^{1+\epsilon}\phantom{m}\text{
for any }~{}~{}\epsilon>0.$

We will also need the completed $L$-function

$\Lambda(s,\pi\times\pi):=L_{\infty}(s,\pi\times\pi)L(s,\pi\times\pi)$

where

$L_{\infty}(s,\pi\times\pi):=\Big{(}({\Gamma}_{\mathbb{R}}(s){\Gamma}_{\mathbb{R}}(s+1)\Big{)}^{2}\Big{(}{\Gamma}_{\mathbb{C}}(s+k-1){\Gamma}_{\mathbb{C}}(s+k-2)\Big{)}^{2}{\Gamma}_{\mathbb{C}}(s+1){\Gamma}_{\mathbb{C}}(s+2k-3)$

(see [DKS14, p.375] or [PSS14, Section 5.2]) with
${\Gamma}_{\mathbb{C}}(s):=2(2\pi)^{-s}{\Gamma}(s)$ and
${\Gamma}_{\mathbb{R}}(s):=\pi^{-s/2}{\Gamma}(s/2)$. This also satisfies the
usual functional equation taking $s\mapsto 1-s$ by the theory of Rankin-
Selberg convolutions on ${\rm GL}_{n}$. By the formula for
$L_{\infty}(s,\pi\times\pi)$, we see that the analytic conductor of
$L(s,\pi\times\pi)$ is

$\mathfrak{q}(s,\pi\times\pi)\ll
q(\pi\times\pi)k^{10}(|s|+3)^{16}\ll_{N}k^{10}(|s|+3)^{16}.$

Hence the convexity bound (see [IK04, p.101]) for $L(s,\pi\times\pi)$ in the
weight/$t$ aspect is

(16)
$|L(1/2+\epsilon+it,\pi\times\pi)|~{}\ll_{N,\epsilon}~{}k^{\frac{5}{2}+\epsilon}(3+|t|)^{4+\epsilon}\phantom{m}\text{
for any }~{}~{}\epsilon>0.$

##  3\. Sign changes of Hecke eigenvalues over primes

In this section we will prove Theorems 1.1.1 and 1.1.2.

###  3.1. First sign change of Hecke eigenvalues of non-CAP forms

In this subsection, using a technique of Jin [Jin19, Theorem 3.1] (also see
Cho and Kim [CK21]) we prove Theorem 1.1.1 for the non-CAP forms.

######  Proposition 3.1.1.

Let $N\geq 1$ be an integer, and let $F\in S_{k}({\Gamma}_{0}^{(2)}(N))^{\rm
T}$ be a Hecke eigenform at all good primes with normalized Hecke eigenvalues
$\lambda(m)$ for $\gcd(m,N)=1$. Assume that $F$ satisfies the Ramanujan
conjecture.666This is automatic if $k\geq 3$ or $F$ is a Yoshida lift. Then
there exists a prime $p$ with $\gcd(p,N)=1$ and

$p\ll_{N,\epsilon}k^{5+\epsilon}$

such that $\lambda(p)<0$.

###### Proof.

Fix some $\pi\in\Pi(F)$. For $\mathrm{Re}(s)>1$, let us define

$L^{N,b}(s,\pi)~{}:=\sum_{m\geq 1,\gcd(m,N)=1\atop m\text{ sq-
free}}\frac{\lambda(m)}{m^{s}}~{}=~{}\prod_{(p,N)=1}\left(1+\frac{\lambda(p)}{p^{s}}\right)$

and

$L^{N,b}(s,\pi\times\pi)~{}:=~{}\sum_{m\geq 1,\gcd(m,N)=1\atop m\text{ sq-
free}}\frac{\lambda(m)^{2}}{m^{s}}~{}=~{}\prod_{(p,N)=1}\left(1+\frac{\lambda(p)^{2}}{p^{s}}\right).$

Note that the Euler product is valid as $\lambda(m)$ is multiplicative. For
$\mathrm{Re}(s)>1$, using (7), (9) and (8), we have

$\displaystyle\frac{L^{N,b}(s,\pi)}{L(s,\pi)}$ $\displaystyle=$
$\displaystyle\prod_{p~{}|~{}N}H_{p}(p^{-s})\times\prod_{(p,N)=1}\left(\left(1+\frac{\lambda(p)}{p^{s}}\right)\prod_{1\leq
i\leq 4}\left(1-\frac{\alpha_{p,i}}{p^{s}}\right)\right)$ $\displaystyle:=$
$\displaystyle\prod_{p~{}|~{}N}H_{p}(p^{-s})\times\prod_{(p,N)=1}\left(1+\frac{a_{2}}{p^{2s}}+\cdots+\frac{a_{5}}{p^{5s}}\right),$

where $H_{p}(X)$ is a polynomial of degree $\leq 4$ with absolutely bounded
coefficients and

$\displaystyle\frac{L^{N,b}(s,\pi\times\pi)}{L(s,\pi\times\pi)}$
$\displaystyle=$
$\displaystyle\prod_{p~{}|~{}N}H_{p}^{(1)}(p^{-s})\times\prod_{(p,N)=1}\left(1+\frac{\lambda(p)^{2}}{p^{s}}\right)\prod_{1\leq
i,j\leq 4}\left(1-\frac{\alpha_{p,i}\alpha_{p,j}}{p^{s}}\right)$
$\displaystyle:=$
$\displaystyle\prod_{p~{}|~{}N}H_{p}^{(1)}(p^{-s})\times\prod_{(p,N)=1}\left(1+\frac{b_{2}}{p^{2s}}+\cdots+\frac{b_{17}}{p^{17s}}\right),$

where $a_{i},b_{j}$ for $i=2,\dots,5;j=2,\dots,17$ are complex numbers which
are bounded by an absolute constant and $H_{p}^{(1)}(X)$ is a polynomial of
degree $\leq 16$ with absolutely bounded coefficients. Hence the Euler
products expressing $\frac{L^{N,b}(s,\pi)}{L(s,\pi)}$ and
$\frac{L^{N,b}(s,\pi\times\pi)}{L(s,\pi\times\pi)}$ are absolutely and
uniformly convergent in any compact set in the region $\mathrm{Re}(s)>1/2$.
Therefore, both $\frac{L^{N,b}(s,\pi)}{L(s,\pi)}$ and
$\frac{L^{N,b}(s,\pi\times\pi)}{L(s,\pi\times\pi)}$ define holomorphic
functions in $\mathrm{Re}(s)>1/2$.

Now, we want to show
$|\frac{L^{N,b}(s,\pi\times\pi)}{L(s,\pi\times\pi)}|\asymp_{N}1$ at $s=1$.
First note that, by Weissauer’s bound, none of the Euler factors of
$\frac{L^{N,b}(s,\pi\times\pi)}{L(s,\pi\times\pi)}$ vanishes at $s=1$ any
prime $p$. Hence $\frac{L^{N,b}(s,\pi\times\pi)}{L(s,\pi\times\pi)}\neq 0$ at
$s=1$. Let $C:=\text{max}_{2\leq i\leq 17}|b_{i}|$. Then $C$ is an absolute
constant. Hence one can easily see from (3.1) that

$\left|\frac{L^{N,b}(1,\pi\times\pi)}{L(1,\pi\times\pi)}\right|\ll_{N}1.$

We also have

$\left|\frac{L^{N,b}(1,\pi\times\pi)}{L(1,\pi\times\pi)}\right|\geq\prod_{p~{}|~{}N}|H_{p}^{(1)}(p^{-1})|\times\prod_{(p,N)=1\atop
p\leq 16C}\left(1-\frac{1}{p}\right)^{16}\times\prod_{(p,N)=1\atop
p>16C}\left(1-\frac{16C}{p^{2}}\right).$

Therefore $|\frac{L^{N,b}(1,\pi\times\pi)}{L(1,\pi\times\pi)}|\asymp_{N}1$.

Let $w$ be a smooth function on $[0,\infty)$ with support in $[0,2]$ and
$w\equiv 1$ on $[0,1]$ and $0\leq w\leq 1$ on $[1,2]$. Also let $\mathaccent
866{w}(s):=\int_{0}^{\infty}w(x)x^{s-1}dx$ be the Mellin transform. Then, for
any real number $X>1$, using Perron type formula [Jin19, Section 2.2] we have

$\sum_{m\geq 1,\gcd(m,N)=1\atop m\text{ sq-
free}}\lambda(m)w\left(\frac{m}{X}\right)~{}=~{}\frac{1}{2\pi
i}\int_{1+\epsilon-i\infty}^{1+\epsilon+i\infty}\mathaccent
866{w}(s)L^{N,b}(s,\pi)x^{s}ds$

and

$\sum_{m\geq 1,\gcd(m,N)=1\atop m\text{ sq-
free}}\lambda(m)^{2}w\left(\frac{m}{X}\right)~{}=~{}\frac{1}{2\pi
i}\int_{1+\epsilon-i\infty}^{1+\epsilon+i\infty}\mathaccent
866{w}(s)L^{N,b}(s,\pi\times\pi)x^{s}ds.$

Since both $\frac{L^{N,b}(s,\pi)}{L(s,\pi)}$ and $L(s,\pi)$ are holomorphic in
the region $\mathrm{Re}(s)>1/2$, the function $L^{N,b}(s,\pi)$ has holomorphic
continuation to $\mathrm{Re}(s)>1/2$. The function $\mathaccent 866{w}(s)$ is
of rapid decay as $\mathrm{Im}(s)\to\infty$ and $L(s,\pi)$ is holomorphic in
$\mathrm{Re}(s)>1/2$ implies that

$\sum_{m\geq 1,\gcd(m,N)=1\atop m\text{ sq-
free}}\lambda(m)w\left(\frac{m}{X}\right)~{}=~{}\frac{1}{2\pi
i}\int_{1/2+\epsilon-i\infty}^{1/2+\epsilon+i\infty}\mathaccent
866{w}(s)L^{N,b}(s,\pi)x^{s}ds.$

The fact $|\frac{L^{N,b}(s,\pi)}{L(s,\pi)}|\ll_{N,\epsilon}~{}1$ in
$\mathrm{Re}(s)~{}\geq~{}1/2+\epsilon$ along with the convexity bound (15) for
$L(s,\pi)$ gives

(18) $\sum_{m\geq 1,(m,N)=1\atop m\text{ sq-
free}}\lambda(m)w\left(\frac{m}{X}\right)~{}\ll_{N,\epsilon}~{}(Xk)^{\frac{1}{2}+\epsilon}.$

Since the poles of $L(s,\pi\times\pi)$ are only at $s=1$, using convexity
bound (16) and arguing as above, one can derive

(19) $\sum_{m\geq 1,(m,N)=1\atop m\text{ sq-
free}}\lambda(m)^{2}w\left(\frac{m}{X}\right)~{}=~{}\text{Res}_{s=1}\left(L^{N,b}(s,\pi\times\pi)\right)\mathaccent
1371{w(1)}X~{}+~{}O\left((Xk^{5})^{\frac{1}{2}+\epsilon}\right).$

If $F$ is general type, $L(s,\pi\times\pi)$ has a simple pole at $s=1$ and
$\text{Res}_{s=1}\left(L^{N,b}(s,\pi\times\pi)\right)~{}\gg_{N}~{}k^{-\epsilon}$
for any $\epsilon>0$ (see [CK21, p.928]). When $F$ is of Yoshida type then
$L(s,\pi\times\pi)$ has a double pole at $s=1$ with residue

$\text{Res}_{s=1}\left(L^{N,b}(s,\pi\times\pi)\right)=2c\gamma\cdot
L(1,\pi_{1}\times\pi_{2})^{2}L(1,{\rm sym}^{2}\pi_{1})L(1,{\rm
sym}^{2}\pi_{2}),$

where $\gamma$ is the Euler constant and $c$ is a constant which depends on
$N$ but not on $k$. Hence, using the results from [Li10], one can easily see
$\text{Res}_{s=1}\left(L^{N,b}(s,\pi\times\pi)\right)\gg_{N}~{}k^{-\epsilon}$
for any $\epsilon>0$. Therefore, in both the cases,
$\text{Res}_{s=1}\left(L^{N,b}(s,\pi\times\pi)\right)\mathaccent
1371{w(1)}X\gg_{N}(Xk^{5})^{\frac{1}{2}+\epsilon}$ when
$X\gg_{N}k^{5+\epsilon}$. Let us assume that $\lambda(m)\geq 0$ for all
square-free $m\leq 2X$ with $\gcd(m,N)=1$. When $X\gg_{N}k^{5+\epsilon}$,
Weissauer’s bound and estimates (18) and (19) imply

$X~{}\ll_{N}~{}\sum_{m\geq 1\atop m\text{ sq-
free}}\lambda(m)^{2}w\left(\frac{m}{X}\right)~{}\ll_{N}~{}X^{\delta}\sum_{m\geq
1\atop m\text{ sq-
free}}\lambda(m)w\left(\frac{m}{X}\right)~{}\ll_{N,\epsilon}~{}X^{\delta}(Xk)^{\frac{1}{2}+\epsilon}$

for any $\delta>0$. This is a contradiction to the assumption
$X\gg_{N,\epsilon}k^{5+\epsilon}$. Since $\lambda(m)$ is multiplicative, hence
there exists a prime $p$ with $\gcd(p,N)=1$ and

$p~{}\ll_{N,\epsilon}~{}k^{5+\epsilon}$

such that $\lambda(p)<0$. ∎

###  3.2. Sign changes of Hecke eigenvalues of non-CAP forms over arithmetic
progressions

We begin by recalling the following well-known result, which is a known
special case of the Selberg orthogonality conjecture.

######  Proposition 3.2.1.

Let $\pi$ and $\pi^{\prime}$ be irreducible unitary cuspidal automorphic
representations of ${\rm GL}_{m}({\mathbb{A}})$ and ${\rm
GL}_{m^{\prime}}({\mathbb{A}})$ where $m$ and $m^{\prime}$ are both less than
or equal to 4. Assume that at least one of $\pi$ and $\pi^{\prime}$ is self
dual. Let $N$ be any positive integer. Then

$\sum_{p\leq X,\
\gcd(p,N)=1}\frac{\lambda_{\pi}(p)\overline{\lambda_{\pi^{\prime}}(p)}}{p}=\begin{cases}\log\log
X+O_{\pi,\pi^{\prime},N}(1)&\text{if }\pi^{\prime}\simeq\pi,\\\
O_{\pi,\pi^{\prime},N}(1)&\text{ otherwise.}\end{cases}$

###### Proof.

See Section 2 of [RS96] or [LWY05, Corollary 1.5]. Note here that since only
finitely many primes divide $N$, the condition $\gcd(p,N)=1$ does not change
the asymptotic behaviour of the sums. ∎

As a corollary of the above we obtain

######  Corollary 3.2.2.

Let $\pi$ be an irreducible unitary cuspidal automorphic self-dual
representation of ${\rm GL}_{m}({\mathbb{A}})$ with $m\leq 4$ and assume that
$\pi$ is not the trivial representation of ${\rm GL}_{1}({\mathbb{A}})$. Let
$a,M$ be positive integers such that $\gcd(aq(\pi),M)=1$ where $q(\pi)$ is the
finite conductor of $\pi$. Let $N$ be any positive integer. Then, for
sufficiently large $X$, we have

$\sum_{p\leq X,(p,N)=1\atop p\equiv a{\,\rm
mod\,}M}\frac{\lambda_{\pi}(p)}{p}~{}=~{}O_{\pi,M,N}(1)\phantom{m}$

and

$\sum_{p\leq X,(p,N)=1\atop p\equiv a{\,\rm
mod\,}M}\frac{\lambda_{\pi}(p)^{2}}{p}~{}=~{}\frac{1}{\varphi(M)}\log\log
X~{}+~{}O_{\pi,M,N}(1)$

###### Proof.

For $\gcd(a,M)=1$, by orthogonality of characters, we have

$\sum_{\chi{\,\rm
mod\,}M}\chi(a)^{-1}\chi(p)~{}=~{}\begin{cases}\varphi(M)&\text{if
}~{}~{}p\equiv a{\,\rm mod\,}M\\\ 0&\text{otherwise}.\end{cases}$

Therefore, we have

$\sum_{p\leq X,(p,N)=1\atop p\equiv a{\,\rm
mod\,}M}\frac{\lambda_{\pi}(p)}{p}~{}=~{}\frac{1}{\varphi(M)}\sum_{\chi{\,\rm
mod\,}M}\chi(a)^{-1}\sum_{p\leq
X\atop(p,NM)=1}\frac{\lambda_{\pi\times\chi}(p)}{p}.$

We apply Proposition 3.2.1 to the inner sum, which gives us the first part of
the proposition.

For the second part, observe that $\pi\not\simeq\pi\times\chi$ unless $\chi$
is trivial. This follows from the fact that the conductors are different:
since there exists a prime $p$ dividing the conductor of $\chi$ but not
dividing $q(\pi)$, we see that the $p$-part of the conductor of $\pi$ is
trivial but the $p$-part of the conductor of $\pi\times\chi$ is non-trivial.
As

$\sum_{p\leq X,(p,N)=1\atop p\equiv a{\,\rm
mod\,}M}\frac{\lambda_{\pi}(p)^{2}}{p}~{}=~{}\frac{1}{\varphi(M)}\sum_{\chi{\,\rm
mod\,}M}\chi(a)^{-1}\sum_{p\leq
X\atop(p,NM)=1}\frac{\lambda_{\pi}(p)\lambda_{\pi\times\chi}(p)}{p},$

we apply Proposition 3.2.1 to the inner sum to complete the proof. ∎

######  Lemma 3.2.3.

Let $F\in S_{k}({\Gamma}_{0}^{(2)}(N))^{\rm T}$ be a Hecke eigenform at all
good primes with normalized Hecke eigenvalues $\lambda(m)$ for $\gcd(m,N)=1$.
Also let $a,M$ be positive integers such that $\gcd(aN,M)=1$. Then, for
sufficiently large $X$, we have

$\sum_{p\leq X,(p,N)=1\atop p\equiv a{\,\rm
mod\,}M}\frac{\lambda(p)}{p}~{}=~{}O_{F,M}(1)\phantom{m}$

and

$\sum_{p\leq X,(p,N)=1\atop p\equiv a{\,\rm
mod\,}M}\frac{\lambda(p)^{2}}{p}~{}=~{}\begin{cases}\frac{1}{\varphi(M)}\log\log
X~{}+~{}O_{F,M}(1)&\text{ if }F\text{ is of general type};\\\
\frac{2}{\varphi(M)}\log\log X~{}+~{}O_{F,M}(1)&\text{ if }F\text{ is of
Yoshida type}.\end{cases}$

###### Proof.

This follows immediately from Proposition 3.2.1, Corollary 3.2.2 and the fact
that

$\lambda(p)~{}=~{}\begin{cases}\lambda_{\Pi}(p)&\text{ if }F\text{ is of
general type};\\\ \lambda_{\pi_{1}}(p)+\lambda_{\pi_{2}}(p)&\text{ if }F\text{
is of Yoshida type}.\end{cases}$

where $\Pi$ is an irreducible unitary cuspidal automorphic self-dual
representation of ${\rm GL}_{4}({\mathbb{A}})$ and $\pi_{1},\pi_{2}$ are non-
isomorphic unitary cuspidal automorphic self-dual representation of ${\rm
GL}_{2}({\mathbb{A}})$. ∎

######  Proposition 3.2.4.

Let $F\in S_{k}({\Gamma}_{0}^{(2)}(N))^{\rm T}$ be a Hecke eigenform at all
good primes with normalized Hecke eigenvalues $\lambda(m)$ for $\gcd(m,N)=1$.
If $k=2$, then assume further that $F$ satisfies the Ramanujan conjecture.
Also let $a,M$ be positive integers such that $\gcd(aN,M)=1$. Then, the
sequence $\\{\lambda(p)\\}_{p\equiv a{\,\rm mod\,}M}$ has infinitely many
positive and infinitely many negative elements.

###### Proof.

We prove the proposition when $F$ is of general type. The case of Yoshida type
follows same argument with obvious modifications. For $0<\delta<1$, let
$y=X^{\delta}$. Without loss of generality, assume that $\lambda(p)>0$ for any
prime $y\leq p\leq X$ and $p\equiv a{\,\rm mod\,}M$. By Weissuaer’s estimate
$|\lambda(p)|\leq 4$. Therefore Lemma 3.2.3 implies

$\frac{1}{\varphi(M)}\log\left(\frac{\log X}{\log
y}\right)~{}+~{}O_{F,M}(1)~{}=~{}\sum_{y\leq p\leq X,(p,N)=1\atop p\equiv
a{\,\rm mod\,}M}\frac{\lambda(p)^{2}}{p}~{}\leq~{}4\sum_{y\leq p\leq
X,(p,N)=1\atop p\equiv a{\,\rm mod\,}M}\frac{\lambda(p)}{p}~{}=~{}O_{F,M}(1).$

This is impossible for sufficiently small $\delta>0$ which completes the
proof. ∎

###  3.3. The case of CAP forms

In the case of CAP forms we can prove a stronger result.

######  Proposition 3.3.1.

Let $F\in S_{k}({\Gamma}_{0}^{(2)}(N))^{\rm CAP}$ be a Hecke eigenform at all
good primes with normalized Hecke eigenvalues $\lambda(m)$ for $\gcd(m,N)=1$.
Assume that $F$ does not lie in $S_{k}^{*}(\Gamma^{(2)}_{0}(N))$. Then there
exists a prime $p$ with $\gcd(p,N)=1$ and

$p~{}\ll_{N}~{}1$

such that $\lambda(p)<0$.

###### Proof.

We go through the various types of $F$ using the terminology of Proposition
2.3.1. First, let $F$ be of Saito–Kurokawa type with $\chi_{0}\neq 1$. For any
prime $\gcd(p,N)=1$, we have

$\lambda(p)=\lambda_{0}(p)+\left(p^{1/2}+p^{-1/2}\right)\chi_{0}(p),$

where $\lambda_{0}(p),\chi_{0}$ are as in Proposition 2.3.1. Since
$|\lambda_{0}(p)|\leq 2$, the sign of $\lambda(p)$ is the same as that of
$\chi_{0}(p)$. Hence the bound for first sign change does not depend on the
weight $k$ but it depends only on $\chi_{0}$ and consequently only on $N$
(since there are only finitely many quadratic Dirichlet characters unramified
away from $N$). Therefore, in this case, there exists prime $p$ with
$\gcd(p,N)=1$ and $p\ll_{N}1$ such that $\lambda(p)<0$.

If $F$ is of Soudry type, then, for any prime $\gcd(p,N)=1$,

$\lambda(p)=\left(p^{1/2}+p^{-1/2}\right)\lambda_{0}(p),$

where $\lambda_{0}(p)$ is as in Proposition 2.3.1. Therefore the bound for the
first sign change is the same as that of $\lambda_{0}(p)$. Recall here that
the $\lambda_{0}(p)$ are the Hecke eigenvalues for a cusp form of ${\rm
GL}_{2}$ of weight $1$ and level bounded by a constant depending only on $N$.
So by Theorem 1.1 of [CK21] there exists prime $p$ with $\gcd(p,N)=1$ and
$p\ll_{N}1$ such that $\lambda_{0}(p)<0$. This implies that $\lambda(p)<0$.

Finally, we assume that $F$ is of Howe–Piatetski-Shapiro type. Hence for any
prime $\gcd(p,N)=1$ we have

$\lambda(p)=\left(p^{1/2}+p^{-1/2}\right)\left(\chi_{1}(p)+\chi_{2}(p)\right),$

where as in Proposition 2.3.1, the Dirichlet characters $\chi_{1},\chi_{2}$
are distinct. Clearly the sign of $\lambda(p)$ is the same as that of
$\chi_{1}(p)+\chi_{2}(p)$ and for each $N$ there are only finitely many
possibilities for $\chi_{1}(p)+\chi_{2}(p)$. It only remains to prove that
given $\chi_{1}$ and $\chi_{2}$ that $\chi_{1}(p)+\chi_{2}(p)$ changes sign
infinitely often. Indeed, we have the following more precise information. Let
$K_{1},K_{2}$ be quadratic fields associated to the quadratic Dirichlet
characters $\chi_{1},\chi_{2}$ respectively. The Dirichlet characters
$\chi_{1}\neq\chi_{2}$ implies $K_{1}\neq K_{2}$. Then, for $\gcd(p,N)=1$ we
have

(20) $\chi_{1}(p)+\chi_{2}(p)~{}=~{}\begin{cases}2&\text{if }p\text{ is split
in both }K_{1}\text{ and }K_{2},\\\ -2&\text{if }p\text{ is inert in
}K_{1}\text{ and }K_{2},\\\ 0&\text{if }p\text{ is split in one field and
inert in other field}.\end{cases}$

Therefore, $\chi_{1}(p)+\chi_{2}(p)$ changes sign infinitely often as $p$ runs
over $\gcd(p,N)=1$. ∎

######  Proposition 3.3.2.

Let $F\in S_{k}({\Gamma}_{0}^{(2)}(N))^{\rm CAP}$ be a Hecke eigenform at all
good primes with normalized Hecke eigenvalues $\lambda(m)$ for $\gcd(m,N)=1$.
Assume that $F$ does not lie in $S_{k}^{*}(\Gamma^{(2)}_{0}(N))$. Then, for
any positive integers $a,M$ with $\gcd(aN,M)=1$, the sequence
$\\{\lambda(p)\\}_{p\equiv a{\,\rm mod\,}M}$ changes sign infinitely often.

###### Proof.

The proof for the various cases follow a similar pattern. We shall give full
details of the proof when $F$ is of non-classical Saito–Kurokawa type and
indicate the main changes in the proof for the Howe–Piatetski-Shapiro and
Soudry types.

Assume that $F\in S_{k}({\Gamma}_{0}^{(2)}(N))$ is of Saito–Kurokawa type but
not of classical Saito–Kurokawa type. Then, by Proposition 2.3.1, there is a
primitive quadratic Dirichlet character $\chi_{0}$ such that

$\lambda(p)=\lambda_{0}(p)+\left(p^{1/2}+p^{-1/2}\right)\chi_{0}(p)$

for all primes $\gcd(p,N)=1$. Here $\lambda_{0}(p)$ is as in Proposition
2.3.1. We note here that sign of $\lambda(p)$ is same as that of $\chi_{0}(p)$
when $\gcd(p,N)=1$. It remains to prove that $\chi_{0}(p)$ changes sign
infinitely often when $p$ runs over $p\equiv a{\,\rm mod\,}M$ for any
$\gcd(aN,M)=1$.

Observe that, for $X$ sufficiently large, we have

$\displaystyle\sum_{p\leq X,(p,N)=1\atop{p\equiv a{\,\rm
mod\,}M\atop\chi_{0}(p)=1}}\frac{1}{p}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{p\leq
X\atop(p,N)=1}\frac{\chi_{0}(p)+1}{p}\times\frac{1}{\varphi(M)}\sum_{\chi{\,\rm
mod\,}M}\chi(a)^{-1}\chi(p)$ $\displaystyle=$
$\displaystyle\frac{1}{2\varphi(M)}\sum_{\chi{\,\rm
mod\,}M}\chi(a)^{-1}\left(\sum_{p\leq
X\atop(p,N)=1}\frac{\chi_{0}(p)\chi(p)}{p}+\sum_{p\leq
X\atop(p,N)=1}\frac{\chi(p)}{p}\right).$

Note that $\chi_{0}\chi\neq 1$ as $\gcd(N,M)=1$. Hence

$\sum_{p\leq X,(p,N)=1\atop{p\equiv a{\,\rm
mod\,}M\atop\chi_{0}(p)=1}}\frac{1}{p}=\frac{1}{2\varphi(M)}\log\log
X+O_{F,M}(1).$

Similarly, using $1-\chi_{0}(p)$ in (3.3) instead of $\chi_{0}(p)+1$, one can
show

$\sum_{p\leq X,(p,N)=1\atop{p\equiv a{\,\rm
mod\,}M\atop\chi_{0}(p)=-1}}\frac{1}{p}=\frac{1}{2\varphi(M)}\log\log
X+O_{F,M}(1).$

This completes the proof in the case when $F$ is a non-classical
Saito–Kurokawa lift.

When $F$ is of Howe–Piatetski-Shapiro type and $\chi_{1},\chi_{2}$ are two
distinct quadratic Dirichlet characters as in Proposition 2.3.1. We need to
show that $\chi_{1}(p)+\chi_{2}(p)$ takes infinitely many positive and
negative values as $p$ runs over $p\equiv a{\,\rm mod\,}M$ when
$\gcd(aN,M)=1$. From (20), it is clear that one can use

$\frac{1}{8}\big{(}\chi_{1}(p)+\chi_{2}(p)+2\big{)}\big{(}\chi_{1}(p)+\chi_{2}(p)\big{)}=\frac{1}{4}\big{(}1+\chi_{1}(p)\chi_{2}(p)+\chi_{1}(p)+\chi_{2}(p)\big{)}$

and

$\frac{1}{8}\big{(}\chi_{1}(p)+\chi_{2}(p)-2\big{)}\big{(}\chi_{1}(p)+\chi_{2}(p)\big{)}=\frac{1}{4}\big{(}1+\chi_{1}(p)\chi_{2}(p)-\chi_{1}(p)-\chi_{2}(p)\big{)}$

as characteristic functions for
$\\{(p,N)=1~{}|~{}\chi_{1}(p)+\chi_{2}(p)>0\\}$ and
$\\{(p,N)=1~{}|~{}\chi_{1}(p)+\chi_{2}(p)<0\\}$ respectively and argue as
above to complete the proof.

Finally, we consider the case when $F$ is of Soudry type. By Proposition
2.3.1, there is a unitary cuspidal automorphic representation $\pi_{0}$ of
${\rm GL}_{2}(\mathbb{A})$ such that

$\lambda(p)=\left(p^{1/2}+p^{-1/2}\right)\lambda_{0}(p)$

for any prime $\gcd(p,N)=1$. Here $\lambda_{0}(p)$ is the normalized Hecke
eigenvalue of $\pi_{0}$ at $p$. Clearly, it is enough to prove the statement
of the proposition for $\lambda_{0}(p)$ which can be done in a similar manner
to the proof of Proposition 3.2.4. We leave the details for the reader. ∎

_Proofs of main results._ Theorem 1.1.1 follows from Propositions 3.1.1 and
3.3.1. Theorem 1.1.2 follows from Propositions 3.2.4 and 3.3.2. ∎

###  3.4. Odds and ends

We explain some of the issues that can occur with Theorem 1.1.2 when
$\gcd(N,M)>1$. If the level $N$ is square free, then Theorem 1.1.2 can be
shown to hold for any positive integers $a$ and $M$ such that $\gcd(a,M)=1$.
In this case there is no need to assume $\gcd(N,M)=1$. However for non-
squarefree levels $N$ there are some subtle issues.

For an immediate example of the kind of issues that can occur, consider the
case when $F$ is of Yoshida type which is coming from two CM forms $f_{1}$ of
weight $2$ and $f_{2}$ of weight $2k-2$ and assume that they both have CM by
the same imaginary quadratic field $K$. Then
$\lambda(p)=\lambda_{1}(p)+\lambda_{2}(p)=0$ for all primes $p$ which are
inert in $K$. Hence there is an arithmetic progression $a{\,\rm mod\,}M$ where
$\gcd(M,N)>1$ such that $\lambda(p)=0$ for all $p\equiv a{\,\rm mod\,}M$.

Furthermore, for the CAP forms of Saito-Kurokawa or Howe–Piatetski-Shapiro
type, one can construct arithmetic progressions (with the modulus $M$ only
divisible by primes dividing the level) such that all Hecke eigenvalues at
primes in the progression are positive.

######  Lemma 3.4.1.

Let $F\in S_{k}({\Gamma}_{0}^{(2)}(N))$ be a Hecke eigenform with normalized
eigenvalues $\lambda(m)$. Also assume that $F$ is either of Saito–Kurokawa
type or of Howe–Piatetski-Shapiro type. Then there exists a positive integer
$M$ which is only divisible by primes dividng $N$ such that $\lambda(p)>0$ for
all $p\equiv 1{\,\rm mod\,}M$.

###### Proof.

Let us first assume that $F\in S_{k}({\Gamma}_{0}^{(2)}(N))$ is of
Saito–Kurokawa type. Then, by Proposition 2.3.1, there is a primitive
Dirichlet character $\chi_{0}{\,\rm mod\,}M$ with $\chi^{2}=1$ and a
representation $\pi_{0}$ of ${\rm GL}_{2}(\mathbb{A})$ with normalized Hecke
eigenvalues $\lambda_{0}(p)$ at primes such that

$\lambda(p)=\lambda_{0}(p)+p^{1/2}\chi_{0}(p)+p^{-1/2}\chi_{0}(p)$

for all primes $\gcd(p,N)=1$. Note that if $p\equiv 1{\,\rm mod\,}M$ then
$\chi_{0}(p)=1$ and hence $\lambda(p)=\lambda_{0}(p)+p^{1/2}+p^{-1/2}>0$ for
all $p\equiv 1{\,\rm mod\,}MN$.

Let $F$ be of Howe–Piatetski-Shapiro type. Proposition 2.3.1 implies that
there are two distinct quadratic Dirichlet characters $\chi_{1},\chi_{2}$ with
conductors $M_{1},M_{2}$ respectively such that

$\lambda(p)=\left(p^{1/2}+p^{-1/2}\right)\chi_{1}(p)+\left(p^{1/2}+p^{-1/2}\right)\chi_{2}(p)$

for all $\gcd(p,N)=1$. Note that if $p\equiv 1{\,\rm mod\,}M_{1}M_{2}$ then
$\chi_{1}(p)=1=\chi_{2}(p)$. Hence $\lambda(p)=2(p^{1/2}+p^{-1/2})>0$ for all
$p\equiv 1{\,\rm mod\,}M_{1}M_{2}N$. ∎

##  4\. Sign changes of radial Fourier coefficients of cusp forms

The main aim of this section is to prove Theorem 1.2.1. We divide the proof
into a few parts.

###  4.1. Relations between Fourier coefficients

We will need the following proposition.

######  Proposition 4.1.1.

Let $F\in S_{k}({\Gamma}_{0}^{(2)}(N))$ be a Hecke eigenform at all good
primes having Fourier coefficients $a(T)~{}(T\in{\mathcal{N}}_{2})$. Then, for
a fundamental matrix $T_{0}\in{\mathcal{N}}_{2}$ and a prime $p$ with
$\gcd(p,N)=1$ such that $p$ is inert in $K={\mathbb{Q}}(\sqrt{-4~{}{\rm
det}~{}T_{0}})$, we have

$a(pT_{0})=a(T_{0})\eta(p)$

where $\eta(p)$ is the $p$-th Hecke eigenvalue of $F$.

###### Proof.

This can be derived from a fundamental identity of Andrianov [And09, Theorem
5.28]. However we give here another proof relying on Bessel models. Let
$d:=-4\ {\rm det}~{}T_{0}$, and $\Lambda$ be a character of the class group
${\rm Cl}_{d}$ of $K:={\mathbb{Q}}(\sqrt{d})$. It is well-known that (see,
e.g., [And74, p.84] or [PSS17, Prop 5.3]) that ${\rm Cl}_{d}$ can be
identified with the set of ${\rm SL}_{2}({\mathbb{Z}})$-equivalence classes of
matrices $S\in{\mathcal{N}}_{2}$ with $-4\ {\rm det}~{}S=d$. Let
$\\{N_{i}\\}_{i=1}^{h}$ be a complete set of representatives of ${\rm Cl}_{d}$
with $N_{1}=T_{0}$. By a special case of Corollary 2 of [Mar21] (which
generalizes Theorem 2.10 of [KST12]) we obtain for each $\gcd(p,N)=1$ the
relation

(22)
$\sum_{i=1}^{h}\Lambda(N_{i})a(pN_{i})=p^{k-\frac{3}{2}}U^{1,0}_{p}(\alpha_{p,1},\alpha_{p,3};K,\Lambda)\sum_{i=1}^{h}\Lambda(N_{i})a(N_{i}),$

where $\alpha_{p,i}$ are the Satake parameters and
$U^{\ell,m}_{p}(-,-;K,\Lambda)$ is the (normalized) Bessel function attached
to a spherical representation, as defined in [KST12, Theorem 2.5]. Using
Sugano’s formula for $U^{1,0}_{p}$ as given in [KST12, (2.3.6)], we have that

(23) $U^{1,0}_{p}(a,b;K,\Lambda)=a+b+a^{-1}+b^{-1}-p^{-1/2}\gamma,$

where for each $p$ coprime to $N$ we have

(24) $\gamma=\begin{cases}0&\text{ if }p\text{ is inert in }K,\\\
\Lambda_{p}(\varpi_{K,p})&\text{ if }p\text{ is ramified in }K,\\\
\Lambda_{p}(\varpi_{K,p})+\overline{\Lambda_{p}(\varpi_{K,p})}&\text{ if
}p\text{ is split in }K.\end{cases}$

By our assumption $p$ is inert in $K$, so we have $\gamma=0$ and thus obtain
$p^{k-\frac{3}{2}}(U^{1,0}_{p}(\alpha_{p,1},\alpha_{p,3}))=p^{k-\frac{3}{2}}(\alpha_{p,1}+\alpha_{p,3}+\alpha_{p,1}^{-1}+\alpha_{p,3}^{-1})=\eta(p)$
and so (22) reduces to

$\sum_{i=1}^{h}\Lambda(N_{i})a(pN_{i})=\eta(p)\sum_{i=1}^{h}\Lambda(N_{i})a(N_{i}).$

Since this is true for any character $\Lambda$ of ${\rm Cl}_{d}$, we use
orthogonality of characters to deduce that $a(pN_{i})=\eta(p)a(N_{i})$ for
each $i$. ∎

Applying Theorem 1.1.2 and Proposition 4.1.1, it is easy to obtain the
following weaker form of Theorem 1.2.1.

######  Corollary 4.1.2.

Let $F\in S_{k}({\Gamma}_{0}^{(2)}(N))$ be a Hecke eigenform at all good
primes having Fourier coefficients $a(T)$. Assume that $F$ does not lie in
$S_{k}^{*}({\Gamma}_{0}^{(2)}(N))$. If $k=2$ and $F$ is not attached to a CAP
representation, then assume further that $F$ satisfies the Ramanujan
conjecture. Then for any fundamental $T_{0}\in{\mathcal{N}}_{2}$ such that
$\gcd(4\det(T_{0}),N)=1$ and $a(T_{0})\neq 0$, there are infinitely many
primes $p$ with $a(pT_{0})>0$ and there are infinitely many primes $p$ such
that $a(pT_{0})<0$.

###### Proof.

Let $\eta(p)$ be the $p$-th Hecke eigenvalue of $F$. Then for any fundamental
$T_{0}$ with $a(T_{0})\neq 0$, Proposition 4.1.1 implies

$a(pT_{0})=a(T_{0})\eta(p)$

for any prime $p$ which is coprime to $N$ and inert in
$K={\mathbb{Q}}(\sqrt{-4{\rm det}~{}T_{0}})$. Since the latter set contains an
arithmetic progression modulo $M=4\det(T_{0})$, we apply Theorem 1.1.2 to
complete the proof. ∎

While Corollary 4.1.2 is an easy consequence of Theorem 1.1.2 and Proposition
4.1.1, we will prove below in Proposition 4.3.1 that one can extend Corollary
4.1.2 to an _arbitrary_ cusp form $F\in
S_{k}^{*}({\Gamma}_{0}^{(2)}(N))^{\perp}$. Note here that if $F\in
S_{k}({\Gamma}_{0}^{(2)}(N))$ is a Hecke eigenform at all good primes that
does not lie in $S_{k}^{*}({\Gamma}_{0}^{(2)}(N))$ then $F$ lies in
$S_{k}^{*}({\Gamma}_{0}^{(2)}(N))^{\perp}$.

The next remark clarifies what is known about the existence of fundamental
matrices $T_{0}$ as above. In particular, if $N=1$ (or if one makes some mild
assumptions on $F$ at the primes dividing odd and squarefree $N$) then there
are lots of $T_{0}\in{\mathcal{N}}_{2}$ such that $\gcd(4\det(T_{0}),N)=1$ and
$a(T_{0})\neq 0$.

######  Remark 4.1.3.

Suppose that $N$ is odd and squarefree and $F\in S_{k}(\Gamma^{(2)}_{0}(N))$
is non-zero with Fourier coefficients $a(T)$. If $N>1$, assume that $F$ is an
eigenform for the $U(p)$ Hecke operator for the finitely many primes $p|N$; we
make no assumptions concerning whether $F$ is a Hecke eigenform at primes not
dividing the level $N$. Then by the results of [JLS21], one knows that given
$\epsilon>0$, for all sufficiently large $X$ there are $\geq X^{1-\epsilon}$
distinct odd squarefree integers $n_{i}\in[X,2X]$ satisfying $\gcd(n_{i},N)=1$
and $-n_{i}\equiv 1{\,\rm mod\,}4$ (in particular $-n_{i}$ is a fundamental
discriminant) such that for each $n_{i}$ as above there is a fundamental
matrix $T_{i}$ with $4{\rm det}~{}T_{i}=n_{i}$ and $a(T_{i})\neq 0$.

We will also need another relation which is specific to the classical
Saito–Kurokawa lifts.

######  Lemma 4.1.4.

Let $F\in S_{k}^{*}({\Gamma}_{0}^{(2)}(N))$ be a Hecke eigenform at all good
primes with normalized Hecke eigenvalues $\lambda(m)$ for $\gcd(m,N)=1$. Let
$\pi_{0}$ be the representation of ${\rm GL}_{2}({\mathbb{A}})$ associated to
$F$ (see case 3 of Proposition 2.3.1) and let $\lambda_{0}(n)$ denote the
normalized Hecke eigenvalue of $\pi_{0}$ at $n$. Let
$a(T)~{}(T\in{\mathcal{N}}_{2})$ be the Fourier coefficients of $F$ and let
$T_{0}$ be a fundamental matrix with the fundamental discriminant $d:=-4{\rm
det}~{}T_{0}$. Then

$a(pT_{0})~{}=~{}a(T_{0})p^{k-1}\left[1-\left(\frac{d}{p}\right)\frac{1}{p}~{}+~{}\frac{\lambda_{0}(p)}{p^{1/2}}\right]$

for any prime $p$.

###### Proof.

Let $\\{N_{i}\\}_{i=1}^{h}$ be as in the proof of Proposition 4.1.1. By a
special case of Lemma 9 of [Mar21] (see also Theorem 5.1 of [PSS17])
$a(N_{i})=a(N_{j})$ and $a(pN_{i})=a(pN_{j})$ for $1\leq i,j\leq h$. So
applying (22) with $\Lambda=1$, we obtain

$ha(pT_{0})~{}=~{}ha(T_{0})\left(p^{k-\frac{3}{2}}\lambda(p)-p^{k-2}(1+\left(\frac{d}{p}\right))\right).$

Since $\lambda(p)=\lambda_{0}(p)+p^{1/2}+p^{-1/2}$ by case 3 of Proposition
2.3.1, we obtain the desired result. ∎

###  4.2. A Selberg orthogonality relation for linear combinations of Hecke
eigenforms

The object of this subsection is to prove the following proposition which may
be viewed as an extension of Lemma 3.2.3 to the case of linear combinations of
eigenforms in $S_{k}^{*}({\Gamma}_{0}^{(2)}(N))^{\perp}$ (including
combinations from different types of Arthur packets).

######  Proposition 4.2.1.

Let $k$ and $N$ be positive integers. Let $F_{1},F_{2},\ldots,F_{m}$ be
elements of $S_{k}^{*}({\Gamma}_{0}^{(2)}(N))^{\perp}$ such that each $F_{i}$
is a Hecke eigenform at all good primes with normalized Hecke eigenvalues
$\lambda_{i}(n)$ for $\gcd(n,N)=1$. Let $r_{1},r_{2},\ldots,r_{m}$ be complex
numbers such that $\sum_{i=1}^{m}r_{i}\neq 0$. Denote

$R=(r_{1},r_{2},\ldots,r_{m};F_{1},F_{2},\ldots,F_{m}).$

For each prime $p$ coprime to $N$, define

$a_{R}(p)=\sum_{i=1}^{m}r_{i}\lambda_{i}(p).$

Then exactly one of the following two cases must occur (depending on $R$).

  1. (1)

Case 1. For $a,M$ positive integers with $\gcd(aN,M)=1$ and for all
sufficiently large $X$, we have

$\sum_{p\leq X,(p,N)=1\atop p\equiv a{\,\rm
mod\,}M}\frac{a_{R}(p)}{p}~{}=~{}O_{R,M}(1)\phantom{m}$

and

$\sum_{p\leq X,(p,N)=1\atop p\equiv a{\,\rm
mod\,}M}\frac{|a_{R}(p)|^{2}}{p}~{}=~{}C_{R,M}\log\log X~{}+~{}O_{R,M}(1)$

for some positive real $C_{R,M}$ depending on $R$ and $M$.

Moreover, in this case, under the additional assumption (needed only for
$k=2$) that each $F_{i}$ above that is of general type satisfies the Ramanujan
conjecture, we have $|a_{R}(p)|\ll_{R}1$ for all primes $p$.

  2. (2)

Case 2. For $a,M$ positive integers with $\gcd(aN,M)=1$ and for all
sufficiently large $X$, we have

$\sum_{p\leq X,(p,N)=1\atop p\equiv a{\,\rm
mod\,}M}\frac{a_{R}(p)}{p^{3/2}}~{}=~{}O_{R,M}(1)\phantom{m}$

and

$\sum_{p\leq X,(p,N)=1\atop p\equiv a{\,\rm
mod\,}M}\frac{|a_{R}(p)|^{2}}{p^{2}}~{}=~{}C_{R,M}\log\log X~{}+~{}O_{R,M}(1)$

for some positive real $C_{R,M}$ depending on $R$ and $M$.

Moreover, in this case, $|a_{R}(p)|\ll_{R}p^{1/2}$ for all primes $p$.

###### Proof.

According to Proposition 2.3.1, a Hecke eigenform $F\in
S_{k}^{*}({\Gamma}_{0}^{(2)}(N))^{\perp}$ at all good primes gives rise to a
certain finite set $\Sigma(F)$ of unitary cuspidal automorphic self-dual
irreducible representations of ${\rm GL}_{1}$, ${\rm GL}_{2}$ or ${\rm
GL}_{4}$ such that the Hecke eigenvalues of $F$ are related to those of the
representations in $\Sigma(F)$ as follows:

  * •

If $F$ is of general type, then we take $\Sigma(F)=\\{\Pi\\}$ where $\Pi$ is
as in Case 1 of Proposition 2.3.1. In this case we trivially have
$\lambda_{F}(p)=\lambda_{\Pi}(p)$.

  * •

If $F$ is of Yoshida type, then we take $\Sigma(F)=\\{\pi_{1},\pi_{2}\\}$
where $\pi_{1},\pi_{2}$ are as in Case 2 of Proposition 2.3.1. In this case we
have $\lambda_{F}(p)=\lambda_{\pi_{1}}(p)+\lambda_{\pi_{2}}(p)$.

  * •

If $F$ is of Saito-Kurokawa type, then we take
$\Sigma(F)=\\{\pi_{0},\chi_{0}\\}$ where $\pi_{0},\chi_{0}$ are as in Case 3
of Proposition 2.3.1. In this case we have
$\lambda_{F}(p)=\lambda_{\pi_{0}}(p)+(p^{1/2}+p^{-1/2})\lambda_{\chi_{0}}(p)$.

  * •

If $F$ is of Soudry type, then we take $\Sigma(F)=\\{\pi_{0}\\}$ where
$\pi_{0}$ is as in Case 4 of Proposition 2.3.1. In this case we have
$\lambda_{F}(p)=(p^{1/2}+p^{-1/2})\lambda_{\pi_{0}}(p)$.

  * •

If $F$ is of Howe–Piatetski-Shapiro type, then we take
$\Sigma(F)=\\{\chi_{1},\chi_{2}\\}$ where $\chi_{1},\chi_{2}$ are as in Case 5
of Proposition 2.3.1. In this case we have
$\lambda_{F}(p)=(p^{1/2}+p^{-1/2})\left(\lambda_{\chi_{1}}(p)+\lambda_{\chi_{2}}(p)\right)$.

Let $\\{\pi_{j}\\}_{1\leq j\leq n}$ denote the set of all unitary cuspidal
automorphic self-dual irreducible representations of ${\rm GL}_{1}$, ${\rm
GL}_{2}$ or ${\rm GL}_{4}$ that occur among the elements of $\Sigma(F)$ as
$F\in S_{k}^{*}({\Gamma}_{0}^{(2)}(N))^{\perp}$ varies over Hecke eigenforms
at all good primes. Note that none of the $\pi_{j}$ equals the trivial
representation. Let $\mu_{j}(p)$ denote the normalized Hecke eigenvalue of
$\pi_{j}$ at a prime $p$. For $1\leq i\leq m$, $1\leq j\leq n$ and primes
$p\nmid N$ we set $c_{i}^{(j)}(p)=0$ if $\pi_{j}\notin\Sigma(F_{i})$ and when
$\pi_{j}\in\Sigma(F_{i})$ we set $c_{i}^{(j)}(p)$ equal to either 1 or
$(p^{1/2}+p^{-1/2})$ (depending on the coefficient of the Hecke eigenvalue as
explicated above) so that for all $1\leq i\leq m$ and all $p\nmid N$ we have

(25) $\lambda_{i}(p)=\sum_{j=1}^{n}c_{i}^{(j)}(p)\mu_{j}(p).$

For $1\leq j\leq n$, set

(26) $R^{(j)}(p)=\sum_{i=1}^{m}r_{i}c_{i}^{(j)}(p).$

This shows that

(27) $R^{(j)}(p)=D_{j}+E_{j}(p^{1/2}+p^{-1/2}),$

where $D_{j},E_{j}$ are complex numbers and independent of $p$.

Using (25), (26) and the definition of $a_{R}(p)$, we see that

(28) $a_{R}(p)=\sum_{j=1}^{n}R^{(j)}(p)\mu_{j}(p)$ (29)
$|a_{R}(p)|^{2}=\sum_{j=1}^{n}|R^{(j)}(p)|^{2}\mu_{j}(p)^{2}+\sum_{1\leq
j_{1}\neq j_{2}\leq
n}R^{(j_{1})}(p)\overline{R^{(j_{2})}(p)}\mu_{j_{1}}(p)\mu_{j_{2}}(p).$

We now claim that the function $p\mapsto R^{(j)}(p)$ cannot be identically 0
for all $j$. Indeed, set $C$ to equal the $m\times n$ matrix
$(c_{i}^{(j)}(p))_{i,j}$ and let $V=(v_{j})_{1\leq j\leq n}$ equal the column
matrix defined by

$v_{j}=\begin{cases}1&\text{ if }\pi_{j}\text{ is a ${\rm GL}_{4}$
representation }\\\ 1/2&\text{ if }\pi_{j}\text{ is a ${\rm GL}_{2}$
representation of weight $\geq$ 2 }\\\ \frac{1}{p^{1/2}+p^{-1/2}}&\text{ if
}\pi_{j}\text{ is a ${\rm GL}_{2}$ representation of weight 1 }\\\
\frac{1}{2(p^{1/2}+p^{-1/2})}&\text{ if }\pi_{j}\text{ is a ${\rm GL}_{1}$
character. }\end{cases}$

Then it is easy to check that

$CV=(1,1,\ldots,1)^{t}.$

Consequently, we have

$\displaystyle(R^{(1)}(p),\ R^{(2)}(p),\ \ldots,R^{(n)}(p))\ V$
$\displaystyle=(r_{1},\ r_{2},\ \ldots,r_{m})CV$
$\displaystyle=r_{1}+r_{2}+\ldots r_{m}$ $\displaystyle\neq 0$

thus showing that the claim is true. Now that we have proved that $R^{(j)}(p)$
cannot be identically 0, using the definition of $R^{(j)}(p)$ we see that
there are two cases.

In the first case (which corresponds to Case 1 of the statement of the
proposition) none of the $R^{(j)}(p)$ actually depend on $p$; i.e, in the
notation of (27), each $E_{j}=0$. In this case, $R^{(j)}(p)\ll_{R}1$ for all
$1\leq j\leq n$ and $p$, and furthermore there exists some $j_{0}$ such that
$R^{(j_{0})}(p)\neq 0$. Therefore using (28), (29) and Corollary 3.2.2, we
conclude the proof. Note that if each $\pi_{j}$ satisfies the Ramanujan
conjecture, then we have $|\mu_{j}(p)|\ll 1$ which gives the required bound on
$a_{R}(p)$ from (29).

In the second case (which corresponds to Case 2 of the statement of the
proposition) at least one of the $R^{(j)}(p)$ depends on $p$. In this case,
using (27), we see that $R^{(j)}(p)\ll_{R}p^{1/2}$ for all $1\leq j\leq n$,
and furthermore there exists some $j_{0}$ such that
$R^{(j_{0})}(p)\asymp_{R}p^{1/2}$. Again using (28), (29) and Corollary 3.2.2,
we conclude the proof. ∎

###  4.3. Sign changes for forms orthogonal to classical Saito-Kurokawa lifts

######  Proposition 4.3.1.

Let $k$ and $N$ be positive integers. Let $F\in
S_{k}^{*}({\Gamma}_{0}^{(2)}(N))^{\perp}$ be a non-zero cusp form having real
Fourier coefficients $a(T)$ and suppose that $a(T_{0})\neq 0$ for some
fundamental $T_{0}\in{\mathcal{N}}_{2}$ with $\gcd(4\det(T_{0}),N)=1$. If
$k=2$, assume that each element of $S_{k}({\Gamma}_{0}^{(2)}(N))$ that is of
general type satisfies the Ramanujan conjecture. Then the sequence
$\\{a(pT_{0})\\}_{p\text{ inert in }{\mathbb{Q}}(\sqrt{-4{\rm det}~{}T_{0}})}$
changes sign infinitely often.

###### Proof.

Let us write $F$ as a finite sum

$F=\sum_{i=1}^{m}F_{i},$

where $F_{i}$’s are Hecke eigenforms at all good places having normalized
Hecke eigenvalues $\lambda_{i}(m)$ (for $m\geq 1,(m,N)=1$) and Fourier
coefficients $a_{i}(T)~{}(T\in{\mathcal{N}}_{2})$. Let us fix a fundamental
$T_{0}\in{\mathcal{N}}_{2}$ such that $a(T_{0})\neq 0$ and put
$K:={\mathbb{Q}}(\sqrt{-4~{}{\rm det}~{}T_{0}})$ and let $P_{K,-}$ denote the
set of primes inert in $K$. Put $M:=4\det(T_{0})$. Fix some $a$ such that
$\gcd(aN,M)=1$ and such that each prime $p\equiv a{\,\rm mod\,}M$ lies in
$P_{K,-}$. It suffices to show that the sequence $\\{a(pT_{0})\\}_{p\equiv
a{\,\rm mod\,}M}$ changes sign infinitely often.

Put $r_{i}=a_{i}(T_{0})$. So $\sum_{i=1}^{m}r_{i}=a(T_{0})\neq 0$. Following
the notation of Proposition 4.2.1, define
$a_{R}(p)=\sum_{i=1}^{m}r_{i}\lambda_{i}(p)$ and let
$R=(r_{1},r_{2},\ldots,r_{m};F_{1},F_{2},\ldots,F_{m})$; note that $R$ depends
only on $F$ and $T_{0}$.

Using Proposition 4.1.1 we have for each $p\equiv a{\,\rm mod\,}M$,

$\frac{a(pT_{0})}{p^{k-1/2}}=\frac{a_{R}(p)}{p},\quad\frac{|a(pT_{0})|^{2}}{p^{2k-2}}=\frac{|a_{R}(p)|^{2}}{p}$

We now consider two cases according to which case of Proposition 4.2.1 we find
ourselves in.

In Case 1, there is a constant $C_{1}$ depending on $F$ and $T_{0}$ such that
for each $p\equiv a{\,\rm mod\,}M$

$\frac{|a(pT_{0})|^{2}}{p^{2k-2}}=\frac{|a_{R}(p)|^{2}}{p}~{}\leq~{}C_{1}\frac{|a_{R}(p)|}{p}=C_{1}\left|\frac{a(pT_{0})}{p^{k-1/2}}\right|,$

and for all large $X$ using Proposition 4.2.1 we have

(30) $\sum_{p\leq X\atop p\equiv a{\,\rm
mod\,}M}\frac{a(pT_{0})}{p^{k-1/2}}~{}=~{}\sum_{p\leq X\atop p\equiv a{\,\rm
mod\,}M}\frac{a_{R}(p)}{p}~{}\ll_{F,T_{0}}~{}1,$

(31) $\displaystyle\sum_{p\leq X\atop p\equiv a{\,\rm
mod\,}M}\frac{|a(pT_{0})|^{2}}{p^{2k-2}}~{}=~{}\sum_{p\leq X\atop p\equiv
a{\,\rm mod\,}M}\frac{|a_{R}(p)|^{2}}{p}=C_{F,T_{0}}\log\log
X+O_{F,T_{0}}(1).$

Now for any $0<\delta<1$, let $y=X^{\delta}$. Assume that $a(pT_{0})$ has the
same sign $\delta\in\\{1,-1\\}$ for any prime $y\leq p\leq X$ and $p\equiv
a{\,\rm mod\,}M$. Therefore (30) and (31) imply

$C_{F,T_{0}}\log\left(\frac{\log X}{\log
y}\right)~{}+~{}O_{F,T_{0}}(1)~{}=~{}\sum_{y\leq p\leq X\atop p\equiv a{\,\rm
mod\,}M}\frac{|a(pT_{0})|^{2}}{p^{2k-2}}~{}\leq~{}C_{1}\delta\sum_{y\leq p\leq
X\atop p\equiv a{\,\rm
mod\,}M}\frac{a(pT_{0})}{p^{k-1/2}}~{}=~{}O_{F,T_{0}}(1).$

This is absurd when $\delta>0$ is sufficiently small. Hence we are done.

In Case 2, for each $p\equiv a{\,\rm mod\,}M$ there is a constant $C_{1}$ such
that

$\frac{|a(pT_{0})|^{2}}{p^{2k-1}}=\frac{|a_{R}(p)|^{2}}{p^{2}}~{}\leq~{}C_{1}\frac{|a_{R}(p)|}{p^{3/2}}=C_{1}\left|\frac{a(pT_{0})}{p^{k}}\right|,$

and for all large $X$ using Proposition 4.2.1 we have

(32) $\sum_{p\leq X\atop p\equiv a{\,\rm
mod\,}M}\frac{a(pT_{0})}{p^{k}}~{}=~{}\sum_{p\leq X\atop p\equiv a{\,\rm
mod\,}M}\frac{a_{R}(p)}{p^{3/2}}~{}\ll_{F,T_{0}}~{}1,$

(33) $\displaystyle\sum_{p\leq X\atop p\equiv a{\,\rm
mod\,}M}\frac{|a(pT_{0})|^{2}}{p^{2k-1}}~{}=~{}\sum_{p\leq X\atop p\equiv
a{\,\rm mod\,}M}\frac{|a_{R}(p)|^{2}}{p^{2}}=C_{F,T_{0}}\log\log
X+O_{F,T_{0}}(1).$

Now for any $0<\delta<1$, let $y=X^{\delta}$. Assume that $a(pT_{0})$ has the
same sign $\delta\in\\{1,-1\\}$ for any prime $y\leq p\leq X$ and $p\equiv
a{\,\rm mod\,}M$. Therefore (32) and (33) imply

$C_{F,T_{0}}\log\left(\frac{\log X}{\log
y}\right)~{}+~{}O_{F,T_{0}}(1)~{}=~{}\sum_{y\leq p\leq X\atop p\equiv a{\,\rm
mod\,}M}\frac{|a(pT_{0})|^{2}}{p^{2k-1}}~{}\leq~{}C_{1}\delta\sum_{y\leq p\leq
X\atop p\equiv a{\,\rm mod\,}M}\frac{a(pT_{0})}{p^{k}}~{}=~{}O_{F,T_{0}}(1).$

This is absurd when $\delta>0$ is sufficiently small. Hence we are done. ∎

###  4.4. Sign changes for forms in the generalized Maass subspace

The next result together with Proposition 4.3.1 completes the proof of Theorem
1.2.1.

######  Proposition 4.4.1.

Let $F\in S_{k}^{*}({\Gamma}_{0}^{(2)}(N))$ be a non-zero cusp form having
real Fourier coefficients $a(T)$ and $a(T_{0})\neq 0$ for some fundamental
$T_{0}$. Then there exists a constant $C>0$ such that the quantities
$a(pT_{0})$ have the same sign for all primes $p>C$.

Proof of Proposition 4.4.1. Since $F\in S_{k}^{*}({\Gamma}_{0}^{(2)}(N))$,
there exist a set of Hecke eigenforms (at all good places)
$F_{1},\dots,F_{\ell}\in S_{k}^{*}({\Gamma}_{0}^{(2)}(N))$ so that

$F=\sum_{i=1}^{\ell}F_{i}.$

We let $a_{i}(T)$ denote the Fourier coefficient of $F_{i}$. Then for any
fundamental $T_{0}\in{\mathcal{N}}_{2}$ with $a(T_{0})\neq 0$ and for any
prime $p$ Lemma 4.1.4 gives

$a_{i}(pT_{0})~{}=~{}a_{i}(T_{0})p^{k-1}\left[1-\left(\frac{d}{p}\right)\frac{1}{p}~{}+~{}\frac{\lambda_{0,i}(p)}{p^{1/2}}\right],$

where $d:=-4{\rm det}~{}T_{0}$. Hence we have

$\displaystyle a(pT_{0})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{\ell}a_{i}(pT_{0})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{\ell}a_{i}(T_{0})p^{k-1}\left[1-\left(\frac{d}{p}\right)\frac{1}{p}~{}+~{}\frac{\lambda_{0,i}(p)}{p^{1/2}}\right]$
$\displaystyle=$ $\displaystyle
a(T_{0})p^{k-1}~{}+~{}O_{F,T_{0}}\left(p^{k-3/2}\right).$

Here in the last line we used $\lambda_{0,i}(p)\ll 1$. The above shows that
for all large $p$, $a(pT_{0})$ has the same sign as $a(T_{0})$ and this
completes the proof. ∎

##  5\. Bounds for Fourier coefficients of cusp forms in the Maass subspace

In this section, we will prove Theorem 1.2.2. We give two proofs, using very
different methods. For the first proof, carried out in Section 5.1, we build
upon some calculations due to Pitale–Schmidt [PS14] to obtain key bounds (see
Proposition 5.1.4) quantifying the growth of the local Bessel functions
associated to certain fixed vectors in representations of ${\rm
GSp}_{4}({\mathbb{Q}}_{p})$. This relates the sizes of general Fourier
coefficients of forms of Saito–Kurokawa type to the sizes of fundamental
Fourier coefficients, and in the latter case the desired bounds easily follow
from bounds for central values of $L$-functions. Using the above method, we
are able to prove a refined form of Theorem 1.2.2 in Theorem 5.1.6 below.

For the second proof, which is classical and carried out in Section 5.2, we
restrict ourselves to Saito-Kurokawa lifts of newforms of squarefree levels;
see the statement of Proposition 5.2.1. To prove Proposition 5.2.1, we follow
a technique of Kohnen [Koh92, Koh93] together with recent developments in the
theory of Fourier–Jacobi expansion.

###  5.1. Bounds for Fourier coefficients using local Bessel functions

For $S={\left[\begin{smallmatrix}a&b/2\\\
b/2&c\end{smallmatrix}\right]}\in{\mathcal{N}}_{2}$ we denote
$L_{S}:=\gcd(a,b,c)$, ${\rm disc}(S):=-4\det(S)=b^{2}-4ac$; we call $L_{S}$
the content of $S$ and call ${\rm disc}(S)$ the discriminant of $S$. For any
$S={\left[\begin{smallmatrix}a&b/2\\\
b/2&c\end{smallmatrix}\right]}\in{\mathcal{N}}_{2}$, the non-zero integer
${\rm disc}(S)$ factors as follows:

${\rm disc}(S)=d_{S}L_{S}^{2}M_{S}^{2}$

where $d_{S}<0$ is a fundamental discriminant, and $M_{S}$ is a positive
integer. Note that $S$ is fundamental if and only if $L_{S}=M_{S}=1$ in the
above factorization. Moreover, any $S\in{\mathcal{N}}_{2}$ is ${\rm
SL}_{2}({\mathbb{Z}})$-equivalent to a matrix

(34) $S^{\prime}:={\left[\begin{smallmatrix}L_{S}&\\\
&L_{S}\end{smallmatrix}\right]}{\left[\begin{smallmatrix}M_{S}&\\\
&1\end{smallmatrix}\right]}S_{c}{\left[\begin{smallmatrix}M_{S}&\\\
&1\end{smallmatrix}\right]}$

where $S_{c}$ is a fundamental matrix of discriminant $d_{S}$; see Proposition
5.3 of [PSS17].

It is clear that the fundamental discriminant $d_{S}$ is uniquely determined
by ${\rm disc}(S)$, and therefore the pair of integers $({\rm disc}(S),L_{S})$
determine $d_{S}$ and $M_{S}$. A key distinguishing property of forms $F$ in
$S_{k}^{*}({\Gamma}_{0}^{(2)}(N))$ — which does not hold in general for
$F\notin S_{k}^{*}({\Gamma}_{0}^{(2)}(N))$ — is that a Fourier coefficient
$a(S)$ of $F$ only depends on ${\rm disc}(S)$ and $L_{S}$ (or equivalently,
only on $d_{S}$, $L_{S}$ and $M_{S}$).

######  Lemma 5.1.1.

Let $F\in S_{k}^{*}({\Gamma}_{0}^{(2)}(N))$ be a non-zero cusp form with
Fourier coefficients $a(T)$. Suppose that $S_{1},S_{2}\in{\mathcal{N}}_{2}$
with ${\rm disc}(S_{1})={\rm disc}(S_{2})$ and $L_{S_{1}}=L_{S_{2}}$. Then
$a(S_{1})=a(S_{2})$.

###### Proof.

We may assume without loss of generality that $F$ gives rise to an irreducible
automorphic representation, and thus is a Hecke eigenform at all good primes.
The lemma was proved in the case $N=1$ in this setting in [PSS17, Theorem 5.1
(i)], and the proof for general $N$ is essentially identical and follows from
[Mar21, Lemma 9] and (34) above. ∎

In the next lemma we collect some facts about the local representations
associated to forms of classical Saito-Kurokawa type.

######  Lemma 5.1.2.

Let $F\in S_{k}^{*}({\Gamma}_{0}^{(2)}(N))$ be a non-zero form in the
generalized Maass subspace such that $F$ is a Hecke eigenform at all good
primes. Let $\pi\in\Pi(F)$ and write $\pi\simeq\otimes_{v}\pi_{v}$. Let
$f_{0}$ be a cuspidal holomorphic newform of weight $2k-2$ for
$\Gamma_{0}(N_{0})$ (see Proposition 2.3.1) whose adelization generates the
automorphic representation $\pi_{0}$ of ${\rm GL}_{2}({\mathbb{A}})$ such that
$L^{N}(s,\pi)=L^{N}(s,\pi_{0})\zeta^{N}(s+1/2)\zeta^{N}(s-1/2)$. Then the
following hold:

  1. (1)

Suppose that $p$ is a prime such that $p\nmid N_{0}$.777Note here that by
Proposition 2.3.1, if $p|N_{0}$ then $p|N$; so the set of $p\nmid N$ is
contained in the set of $p\nmid N_{0}$. Then
$\pi_{p}\simeq\chi_{p}\mathbf{1}_{{\rm GL}(2)}\rtimes\chi_{p}^{-1}$ is a
representation of type IIb in the notation of Table A.15 of [RS07], where
$\chi_{p}$ is an unramified unitary character of ${\mathbb{Q}}_{p}^{\times}$.

  2. (2)

Suppose that $p$ is a prime such that $p|N_{0}$ and $v_{p}(N)=1$. Then we have
$v_{p}(N_{0})=1$, and moreover either $\pi_{p}\simeq\tau(T,\nu^{-1/2})$ is of
type VIb in the notation of Table A.15 of [RS07], or $\pi_{p}\simeq
L((\nu^{1/2}\xi_{p}\mathrm{St}_{{\rm GL}(2)},\nu^{-1/2})$ is of type Vb in the
notation of Table A.15 of [RS07], where $\xi_{p}$ is the unramified non-
trivial quadratic character of ${\mathbb{Q}}_{p}^{\times}$. The former case
above occurs iff $\pi_{0,p}$ is the Steinberg representation, and the latter
case occurs iff $\pi_{0,p}$ is the unramified quadratic twist of the Steinberg
representation.

Furthermore, if $N$ is squarefree, then $F$ gives rise to an irreducible
automorphic representation. Moreover, if $N$ is squarefree then $F$ is a
newform (in the sense of Section 3.2 of [DPSS20]) if and only if $N_{0}=N$.

###### Proof.

By definition, the global Arthur parameter of $\pi$ is equal to
$(\pi_{0}\boxtimes 1)\boxplus(1\boxtimes\nu(2))$; see the notation and
explanation in Section 1 of [Sch20] and note that the quadratic character
$\sigma$ (which equals the character $\chi_{0}$ in the notation of part 3 of
Proposition 2.3.1)) is trivial in the present case. Now the possibilities for
$\pi_{p}$ can be read off from Table 2 of [Sch20]. In particular, if $p$ is a
prime such that such that $p\nmid N_{0}$, then $\pi_{0,p}$ is an unramified
principal series of the form $\chi_{p}\times\chi_{p}^{-1}$, in which case
Table 2 of [Sch20] asserts that $\pi_{p}\simeq\chi_{p}\mathbf{1}_{{\rm
GL}(2)}\rtimes\chi_{p}^{-1}$.

Next suppose $p|N_{0}$. Hence $p|N$ and so $\pi_{p}$ has a vector fixed by
$K_{0,p}(N)$, the group defined in (13). Looking at Table 2 of [Sch20] we see
that $\pi_{p}$ is one of the types Vb, Va*, VIb, VIc, XIb or XIa*, and of
these only Vb and VIb has a vector fixed by $K_{0,p}(N)$ when $v_{p}(N)=1$ (as
can be read off from Table A.15 of [RS07]). So $\pi_{p}$ is either of Type VIb
or Type Vb. Again from Table 2 of [Sch20], we see that in the former case
$\pi_{0,p}$ is the Steinberg representation, and in the latter case
$\pi_{0,p}$ is the unramified quadratic twist of the Steinberg representation.
Clearly in both cases, the conductor exponent of $\pi_{0,p}$ equals 1, and
hence $v_{p}(N_{0})=1$.

Finally, suppose that $N$ is squarefree. By the previous parts we have that
$N_{0}$ is also squarefree and we also know that $N_{0}|N$. Furthermore, we
know exactly the local components $\pi_{p}$ at each place. It follows that any
two representations in $\pi(F)$ are isomorphic, and consequently, $F$ gives
rise to an irreducible automorphic representation. If $N_{0}=N$, then for each
prime $p|N$ (as well as for each prime $p\nmid N)$, $\pi_{p}$ has a
1-dimensional space of $K_{0,p}(N)$-fixed vectors as follows from the previous
parts and Table A.15 of [RS07]; consequently, $F$ is a newform. Conversely, if
$F$ is a newform then $\pi_{p}$ cannot be of Type IIb at any $p|N$, because
the Type IIb representation being spherical does not contain a local newvector
with respect to $K_{0,p}(N)$ (see Theorem 2.3.1 of [Sch05]). Therefore by the
previous parts, $N_{0}=N$.

This concludes the proof. ∎

######  Remark 5.1.3.

If $N$ is not squarefree, then there are 4 further possibilities for $\pi_{p}$
beyond the Types IIb, Vb and VIb mentioned in Lemma 5.1.2, namely the types
Va*, VIc, XIb and XIa*.

Given a prime $p$ and non-negative integers $\ell,m$, define the matrix

$h_{p}(\ell,m)=\left[\begin{smallmatrix}p^{\ell+2m}&&&\\\ &p^{\ell+m}&&\\\
&&1&\\\ &&&p^{m}\end{smallmatrix}\right]$

We also need the subgroup $P_{1,p}\subset{\rm GSp}_{4}({\mathbb{Z}}_{p})$
which is defined as $P_{1,p}:=K_{0,p}(p)$ where $K_{0,p}(N)$ is defined in
(13).

Consider an irreducible admissible representation $\pi$ of ${\rm
GSp}_{4}({\mathbb{Q}}_{p})$ that admits a local Bessel model with respect to a
standard additive character $\theta$ (see Section 2 of [PSS17]) and the
trivial character $\Lambda=\mathbf{1}$ of a quadratic extension $L$ of
${\mathbb{Q}}_{p}$ ($L$ equals either ${\mathbb{Q}}_{p}\oplus{\mathbb{Q}}_{p}$
or a quadratic field extension of ${\mathbb{Q}}_{p}$). We fix an isomorphism
$w\mapsto B_{w}$ (which is uniquely determined up to multiples) of $\pi$ with
its $(\mathbf{1},\theta)$-Bessel model. This associates to each vector $w\in
V_{\pi}$ a function $B_{w}$ on ${\rm GSp}_{4}({\mathbb{Q}}_{p})$ and we are
interested in bounding $B_{w}(h_{p}(\ell,m))$ as a function of $\ell$ and $m$.
The heart of our method is the next proposition, which quantifies the growth
of the functions $B_{w}(h_{p}(\ell,m))$ associated to suitable vectors $w$ in
local representations of Saito-Kurokawa type.

######  Proposition 5.1.4.

  1. (1)

Let $\pi\simeq\chi\mathbf{1}_{{\rm GL}(2)}\rtimes\chi^{-1}$ be a
representation of ${\rm GSp}_{4}({\mathbb{Q}}_{p})$ of type IIb in the
notation of Table A.15 of [RS07], where $\chi$ is an unramified unitary
character of ${\mathbb{Q}}_{p}^{\times}$. Let $w_{0}$ be the (unique up to
multiples) spherical function in the space of $\pi$. Then $B_{w_{0}}(1)\neq 0$
and for each pair of non-negative integers $\ell,m$, we have the bound

$|B_{w_{0}}(h_{p}(\ell,m))|<(\ell+1)(2\ell+2m+1)p^{-\ell-\frac{3m}{2}}|B_{w_{0}}(1)|.$

Moreover, the space of $P_{1,p}$-fixed vectors in $\pi$ has a basis
$\\{w_{0},w_{1},w_{2}\\}$ where $w_{0}$ is as above, and for $i=1,2$ we have
$B_{w_{i}}(1)\neq 0$ and

$|B_{w_{i}}(h_{p}(\ell,m))|=p^{-\frac{3}{2}\ell-\frac{3}{2}m}|B_{w_{i}}(1)|.$

  2. (2)

Let $\pi$ be isomorphic to one of the following representations of ${\rm
GSp}_{4}({\mathbb{Q}}_{p})$:

     * •

The representation $\tau(T,\nu^{-1/2})$ of type VIb in the notation of Table
A.15 of [RS07],

     * •

The representation $L((\nu^{1/2}\xi\mathrm{St}_{{\rm GL}(2)},\nu^{-1/2})$ of
type Vb in the notation of Table A.15 of [RS07] where $\xi$ is the unramified
non-trivial quadratic character of ${\mathbb{Q}}_{p}^{\times}$.

Suppose that $\pi$ admits a $(\mathbf{1},\theta)$ Bessel model and let $w$ be
a non-zero vector in the 1-dimensional space of $P_{1,p}$-fixed vectors in
$\pi$. Then $B_{w}(1)\neq 0$ and for each pair of non-negative integers
$\ell,m$, we have

$B_{w}(h_{p}(\ell,m))=\pm p^{-2\ell-2m}B_{w}(1).$

Above, the sign in $\pm$ equals -1 if and only if $\pi$ is of Type Vb and
$\ell+m$ is odd.

###### Proof.

Suppose that $\pi\simeq\chi\mathbf{1}_{{\rm GL}(2)}\rtimes\chi^{-1}$ is a
representation of ${\rm GSp}_{4}({\mathbb{Q}}_{p})$ of type IIb where $\chi$
is an unramified unitary character of ${\mathbb{Q}}_{p}^{\times}$, and put
$\alpha=\chi(\varpi)$ where $\varpi$ is a uniformizer in ${\mathbb{Q}}_{p}$.
The fact that $B_{w_{0}}(1)\neq 0$ is a consequence of Sugano’s formula
[Sug85]; see also Section 2 of [PSS17]. We may therefore without loss of
generality assume that $B_{w_{0}}(1)=1$. Now by [PSS17, (12)], we obtain that
$|B_{w_{0}}(h_{p}(0,m))|<(2m+1)p^{-3m/2}$ (note that the Satake parameters
$\alpha^{\pm}$ have absolute value 1) for all $m$. Now using [PSS17, (10)] and
the above bound, we obtain

$|B_{w_{0}}(h_{p}(\ell,m))|<\sum_{i=0}^{\ell}(2\ell+2m-2i+1)p^{-\frac{3\ell}{2}-\frac{3m}{2}+\frac{i}{2}}<(\ell+1)(2\ell+2m+1)p^{-\ell-\frac{3m}{2}}.$

Moreover, the space of $P_{1,p}$-fixed vectors in $\pi$ is 3-dimensional and
has a basis $\\{w_{1},w_{2},w_{3}\\}$ consisting of eigenvectors for the
commuting Hecke operators $T_{1,0}$ and $T_{0,1}$ with eigenvalues $(\alpha
p^{3/2},\alpha(p+1)p^{3/2})$, $(\alpha^{-1}p^{3/2},\alpha^{-1}(p+1)p^{3/2})$,
$(p^{2},(\alpha+\alpha^{-1})p^{5/2})$ as calculated in Table 3 of [PS]. Since
$w_{1}+w_{2}+w_{3}=w_{0}$, it follows that $\\{w_{0},w_{1},w_{2}\\}$ is a
basis for the space of $P_{1,p}$-fixed vectors in $\pi$. Now by Proposition
6.1 of [PS14] it follows that $B_{w_{i}}(h_{p}(\ell,m))=\alpha^{\pm\ell\pm
m}p^{-\frac{3\ell}{2}-\frac{3m}{2}}$ which completes the proof of the first
part of the proposition.

Finally, let $\pi$ be either isomorphic to either $\tau(T,\nu^{-1/2})$ or
$L((\nu^{1/2}\xi\mathrm{St}_{{\rm GL}(2)},\nu^{-1/2})$. Using Table 3 of
[PS14], we see that $w$ is an eigenvector for the operators $T_{1,0}$ and
$T_{0,1}$ with pair of eigenvalues $(\epsilon p,\epsilon p(p+1))$ where
$\epsilon=1$ for $\tau(T,\nu^{-1/2})$ and $\epsilon=-1$ for
$L((\nu^{1/2}\xi\mathrm{St}_{{\rm GL}(2)},\nu^{-1/2})$. Now by Proposition 6.1
of [PS14] (noting that $m_{0}=0$ in our case) it follows that
$B_{w}(h_{p}(\ell,m))=\epsilon^{-\ell-m}p^{-2\ell-2m}B_{w}(1).$ The fact that
$B_{w}(1)\neq 0$ follows from Theorem 8.2 of [PS14]. ∎

The next proposition bounds the fundamental Fourier coefficients of cusp forms
in the generalized Maass subspace.

######  Proposition 5.1.5.

Let $F\in S_{k}^{*}({\Gamma}_{0}^{(2)}(N))$ be a non-zero cusp form with
Fourier coefficients $a(T)$. For fundamental $T\in{\mathcal{N}}_{2}$, we have

$|a(T)|~{}\ll_{F,\epsilon}\begin{cases}({\rm
det}~{}T)^{\frac{k}{2}-\frac{7}{12}+\epsilon}&\text{ unconditionally,}\\\
({\rm det}~{}T)^{\frac{k}{2}-\frac{3}{4}+\epsilon}&\text{ assuming the
generalized Lindel\"{o}f hypothesis.}\end{cases}$

###### Proof.

We may assume without loss of generality that $F$ gives rise to an irreducible
representation $\pi\simeq\pi_{F}$ (since any $F\in
S_{k}^{*}({\Gamma}_{0}^{(2)}(N))$ may be written as a sum of such forms). By
Proposition 2.3.1 there exists a representation $\pi_{0}$ of ${\rm
GL}_{2}({\mathbb{A}})$ such that

$L^{N}(s,\pi)=L^{N}(s,\pi_{0})\zeta^{N}(s+1/2)\zeta^{N}(s-1/2).$

Let $T$ be fundamental and put $d=-4\det(T)$. Then by Proposition 5.13 of
[JLS21], we have

$|a(T)|~{}\ll_{F}~{}({\rm
det}~{}T)^{\frac{k}{2}-\frac{3}{4}}L(\tfrac{1}{2},\pi_{0}\times\chi_{d})^{1/2}.$

By [PY19] we have
$L(\tfrac{1}{2},\pi_{0}\times\chi_{d})\ll_{\pi_{0},\epsilon}|d|^{1/3+\epsilon}$
and conditionally on the generalized Lindelöf hypothesis we have
$L(\tfrac{1}{2},\pi_{0}\times\chi_{d})\ll_{\pi_{0},\epsilon}|d|^{\epsilon}$.
The result follows. ∎

We can now prove the main theorem of this section.

######  Theorem 5.1.6.

Let $k$ and $N$ be positive integers and let $F\in
S_{k}^{*}({\Gamma}_{0}^{(2)}(N))$ be a non-zero cusp form with Fourier
coefficients $a(T)$. Write $N=N_{1}N_{2}$ where $N_{1}$ is squarefree and
$\gcd(N_{1},N_{2})=1.$ Then

$|a(T)|\ll_{F,\epsilon}L_{T}^{k-1+\epsilon}M_{T}^{k-3/2+\epsilon}|d_{T}|^{\frac{k}{2}-\frac{7}{12}+\epsilon}$

for all $T\in{\mathcal{N}}_{2}$ satisfying

(35) $\gcd(L_{T}M_{T},N_{2})=1.$

Furthermore, if we assume the generalized Lindelöf hypothesis, then we have

$|a(T)|\ll_{F,\epsilon}L_{T}^{k-1+\epsilon}M_{T}^{k-3/2+\epsilon}|d_{T}|^{\frac{k}{2}-\frac{3}{4}+\epsilon}$

for all $T\in{\mathcal{N}}_{2}$ satisfying (35).

###### Proof.

We may assume without loss of generality that $F$ has the following
properties:

  * •

$F$ gives rise to an irreducible representation $\pi\simeq\otimes_{v}\pi_{v}$,
and therefore (by Lemma 2.2.1) $F$ is a Hecke eigenform at all good primes.

  * •

The adelization $\phi_{F}$ of $F$ corresponds to a factorizable vector
$\phi_{F}=\otimes_{v}\phi_{v}$ in the space of $\pi$.

  * •

If $p|N_{1}$ and $\pi_{p}=\chi\mathbf{1}_{{\rm GL}(2)}\rtimes\chi^{-1}$ is of
type IIb, then $\phi_{p}$ equals one of the vectors $w_{0},w_{1},w_{2}$ in the
notation of Proposition 5.1.4.

Note that the set of $F$ satisfying the above properties linearly generate
$S_{k}^{*}({\Gamma}_{0}^{(2)}(N))$. Note also (by Lemma 5.1.2 that if
$p|N_{1}$ and $\pi_{p}$ is not of Type IIb, then $\pi_{p}$ must be of Type Vb
or VIb and $\phi_{p}$ must correspond to a (unique up to multiples)
$P_{1,p}$-fixed vector in the space of $\pi_{p}$. Finally, Lemma 5.1.2 also
tells us that at each prime $p\nmid N$, $\pi_{p}$ is of Type IIb.

Now, using essentially the same argument as in Corollary 2 of [Mar21] (with
$\Lambda=1$) together with Lemma 5.1.1, we see that for $T$ as in the
statement of the Theorem

$a(T)\prod_{p|{L_{T}M_{T}}}B_{\phi_{p}}(1)=(L_{T}M_{T})^{k}a(S_{T})\prod_{p|{L_{T}M_{T}}}B_{\phi_{p}}(h_{p}(\ell_{p},m_{p})),$

where we write $L_{T}=\prod_{p}p^{\ell_{p}}$, $M_{T}=\prod_{p}p^{m_{p}}$, and
$S_{T}$ is any matrix of discriminant $d_{T}$. Above, we have used that
$\gcd(L_{T}M_{T},N_{2})=1$ and consequently, $B_{\phi_{p}}(1)\neq 0$ for each
$p|L_{T}M_{T}$ by the remarks at the beginning of this proof together with
Proposition 5.1.4. Now, by Proposition 5.1.4,
$\prod_{p|{L_{T}M_{T}}}|B_{\phi_{p}}(h_{p}(\ell_{p},m_{p}))|\ll_{\epsilon}L_{T}^{-1+\epsilon}M_{T}^{-3/2+\epsilon}\prod_{p|{L_{T}M_{T}}}|B_{\phi_{p}}(1)|$.
The result now follows from Proposition 5.1.5. ∎

In the special case that $N$ is squarefree and $F\in
S_{k}^{*}({\Gamma}_{0}^{(2)}(N))$ is a newform (in the sense of [DPSS20,
Section 3.2]) we have an even sharper result.

######  Proposition 5.1.7.

Let $N$ be a squarefree integer. Let $F\in S_{k}^{*}({\Gamma}_{0}^{(2)}(N))$
be a non-zero cusp form with Fourier coefficients $a(T)$, and suppose that $F$
is a newform (in the sense of [DPSS20, Section 3.2]). Then

$|a(T)|\ll_{F,\epsilon}\frac{L_{T}^{k-1+\epsilon}M_{T}^{k-3/2+\epsilon}}{\gcd(L_{T},N^{\infty})(\gcd(M_{T},N^{\infty}))^{1/2}}|d_{T}|^{\frac{k}{2}-\frac{7}{12}+\epsilon}$

for all $T\in{\mathcal{N}}_{2}$, and if we assume the generalized Lindelöf
hypothesis, then the exponent of $|d_{T}|$ can be taken to be
$\frac{k}{2}-\frac{3}{4}+\epsilon$.

###### Proof.

The proof is essentially the same as Theorem 5.1.6 with the additional
information that in this case, we know that $\pi_{p}$ is of Type Vb or Type
VIb for all $p|N$ (see Lemma 5.1.2. The sharper bound results from the fact
that we are always in the situation of part (2) of Proposition 5.1.4 for all
$p|N$. ∎

###### Proof of Theorem 1.2.2.

Note that if $T\in{\mathcal{N}}_{2}$ has the property that $\gcd(4\det(T),N)$
is squarefree, then we must have $\gcd(L_{T}M_{T},N_{2})=1$. Now the result
follows from the observation that

$L_{T}^{k-1}M_{T}^{k-3/2}|d_{T}|^{\frac{k}{2}-\frac{7}{12}}=\frac{(4\det(T))^{\frac{k-1}{2}}}{M_{T}^{\frac{1}{2}}|d_{T}|^{\frac{1}{12}}}\ll_{k}\det(T))^{\frac{k-1}{2}}.$

∎

###  5.2. Bounds for Fourier coefficients using Fourier–Jacobi expansions

The main result of this subsection is the following.

######  Proposition 5.2.1.

Let $k\geq 4$ be an even integer and $N$ be an odd square-free positive
integer. Also let $f\in S_{2k-2}^{new}(\Gamma_{0}(N))$ be a newform and $F\in
S_{k}(\Gamma_{0}^{(2)}(N))$ be its Saito-Kurokawa lift as described in [AB15].
Assume that $a(T)$ is the $T$-th Fourier coefficient of $F$. Then, for any
$\epsilon>0$, we have

(36) $|a(T)|~{}\ll_{F,\epsilon}~{}(\det T)^{\frac{k-1}{2}+\epsilon}.$

The statement of the above proposition refers to the work of Agarwal and Brown
[AB15] who, building upon work of Ibukiyama [Ibu12], wrote down a classical
construction of Saito-Kurokawa lifts obtained from elliptic Hecke eigenforms
of odd squarefree level $N$, and explicated some key properties of these
lifts. In the next lemma, we show that the Saito-Kurokawa lifts constructed by
Agarwal–Brown coincide with the newforms in $S_{k}^{*}({\Gamma}_{0}^{(2)}(N))$
in our sense.

######  Lemma 5.2.2.

Let $k\geq 2$ be even, let $N$ be an odd squarefree integer, and let $f\in
S_{2k-2}^{new}(\Gamma_{0}(N))$ be a newform888Recall that a newform is a Hecke
eigenform lying in the newspace.; let $\pi_{f}$ be the automorphic
representation of ${\rm GL}_{2}({\mathbb{A}})$ generated by $f$. Let $F\in
S_{k}(\Gamma_{0}^{(2)}(N))$ be the Saito-Kurokawa lift of $f$ as described in
[AB15].

Then $F\in S_{k}^{*}({\Gamma}_{0}^{(2)}(N))$ and satisfies
$L^{N}(s,\pi_{F})=L^{N}(s,\pi_{f})\zeta^{N}(s+1/2)\zeta^{N}(s-1/2)$.
Furthermore, $F$ is a newform in the sense of [DPSS20, Section 3.2]. Moreover,
if $M$ is an integer and $G$ is a newform in
$S_{k}^{*}({\Gamma}_{0}^{(2)}(M))$ satisfying
$L^{M}(s,\pi_{G})=L^{M}(s,\pi_{f})\zeta^{M}(s+1/2)\zeta^{M}(s-1/2)$ then $M=N$
and $G=cF$ for some non-zero complex number $c$.

###### Proof.

Let $\pi\in\Pi(F)$. The fact that
$L^{N}(s,\pi)=L^{N}(s,\pi_{f})\zeta^{N}(s+1/2)\zeta^{N}(s-1/2)$ follows from
Theorem 3.5 of [AB15] (observe that our normalization of the $L$-functions
differ from theirs). So by definition $F\in S_{k}^{*}({\Gamma}_{0}^{(2)}(N))$,
and since $N$ is squarefree, Lemma 5.1.2 tells us that $\pi=\pi_{F}$ is
irreducible and $F$ is a newform. Next, let $G$ be as in the statement of the
lemma. Since $G$ is a newform, Lemma 5.1.2 tells us that $M=N$. Let
$\pi^{\prime}$ be the representation generated by the adelization of $G$.
Applying Lemma 5.1.2 to $F$ and $G$, we see that the local components of $\pi$
and $\pi^{\prime}$ are isomorphic at all finite primes, and hence at all
places (since they are clearly isomorphic at infinity). It follows that the
spaces of $\pi$ and $\pi^{\prime}$ coincide. Since these representations have
a 1-dimensional space of $K_{0,p}(N)$-fixed vectors at each prime $p$, it
follows that the adelizations of $F$ and $G$ are multiples of each other.
Hence $F$ and $G$ are multiples of each other. ∎

The rest of the subsection is devoted to the proof of Proposition 5.2.1. We
will assume throughout that $k\geq 4$ is an even integer and $N$ is an odd
squarefree integer. For any $F\in S_{k}({\Gamma}_{0}^{(2)}(N))$, the
Fourier–Jacobi expansion of $F$ is given by

$F(Z)~{}:=~{}\sum_{m=1}^{\infty}\phi_{m}(\tau,z)e^{2\pi im\tau^{\prime}},$

where $Z=\begin{pmatrix}\tau&z\\\
z&\tau^{\prime}\end{pmatrix}\in{\mathcal{H}}_{2}$ with
$\tau,\tau^{\prime}\in{\mathcal{H}}$ and $z\in{\mathbb{C}}$. It is well-known
[EZ85, GS17] that $\phi_{m}\in J_{k,m}^{cusp}(N)$, where $J_{k,m}^{cusp}(N)$
denotes the space of Jacobi cusp form of weight $k$, index $m$ for the group
$\Gamma^{J}(N):=\Gamma_{0}(N)\ltimes{\mathbb{Z}}^{2}$. The following result of
Bringmann [Bri04, Theorem 3.41] (see also Kohnen [Koh92, Koh93]) on the size
of Fourier coefficients of a Jacobi cusp form is crucial for us.

######  Proposition 5.2.3 ([Bri04, Theorem 3.41]).

Let $\phi\in J_{k,m}^{cusp}(N)$ with Fourier coefficients $c(n,r)$. Then

$|c(n,r)|~{}\ll_{\epsilon,k}~{}\left(m^{3/4}+|D|^{1+\epsilon}\right)^{\frac{1}{2}}\cdot\frac{|D|^{k/2-1}}{m^{\frac{k-1}{2}}}\cdot\parallel\phi\parallel,$

where $\parallel\phi\parallel$ denotes the Petersson norm of $\phi$ which is
defined by

$\parallel\phi\parallel^{2}~{}=~{}\langle\phi,\phi\rangle:=\frac{1}{[{\Gamma}^{J}(1):{\Gamma}^{J}(N)]}\int_{\Gamma^{J}(N)\backslash{\mathcal{H}}\times{\mathbb{C}}}|\phi(\tau,z)|^{2}v^{k-3}e^{-4\pi
my^{2}/v}dudvdxdy,$

with $\tau=u+iv\in{\mathcal{H}}$ and $z=x+iy\in{\mathbb{C}}$ and
$D:=r^{2}-4mn$.

We need the following proposition whose proof is similar to Kohnen and
Sengupta [KS17, Theorem 2] (also see [KP21, Section 6]) and relies on a result
of Agarwal–Brown [AB15] (see also [MRV93, MR00, MR02]).

######  Proposition 5.2.4.

Let $k$ be a positive even integer and $N$ be odd square-free positive
integer. Also let $f\in S_{2k-2}^{new}(\Gamma_{0}(N))$ be a newform and $F\in
S_{k}(\Gamma_{0}^{(2)}(N))$ be its Saito-Kurokawa lift as described in [AB15].
Assume that $F$ has Fourier–Jacobi coefficients
$\\{\phi_{m}\\}_{m\in{\mathbb{N}}}$. Then, for any $\epsilon>0$, we have

(37) $\langle\phi_{m},\phi_{m}\rangle\ll_{F,\epsilon}m^{k-1+\epsilon}.$

###### Proof.

For any positive integer $m$, [AB15, Proposition 4.6 and p.659] implies

(38)
$\frac{\langle\phi_{m},\phi_{m}\rangle}{\langle\phi_{1},\phi_{1}\rangle}~{}=\sum_{d~{}|~{}m,\atop(d,N)=1}\alpha(d)d^{k-2}\lambda_{f}(m/d),$

where $\lambda_{f}(m)$ is the Hecke eigenvalue corresponding to $f$ and hence
by Deligne’s bound we have $|\lambda_{f}(m)|\leq m^{k-3/2}\sigma_{0}(m)$. Here
$\sigma_{0}(m)$ denotes the number of positive divisors of $m$. Further,
$\alpha(d)$ is given by

$\alpha(d):=d\prod_{p|d}\left(1+\frac{1}{p}\right)\leq d\sigma_{0}(d).$

Therefore, one can easily see from (38) that

$\frac{\langle\phi_{m},\phi_{m}\rangle}{\langle\phi_{1},\phi_{1}\rangle}\leq
m^{k-3/2}\sum_{d~{}|~{}m,\atop(d,N)=1}d^{1/2}\sigma_{0}(d)\sigma_{0}(m/d)\ll_{\epsilon}m^{k-1+\epsilon}\sum_{d~{}|~{}m,\atop(d,N)=1}1\ll_{\epsilon}m^{k-1+\epsilon}.$

This completes the proof. ∎

###### Proof of Prop. 5.2.1.

The rest of the proof is similar to that of Kohnen [Koh92], mutatis mutandis.
We sketch the proof here for convenience of reader. Since both sides of (36)
are invariant under $T\mapsto U^{t}TU$ for any $U\in{\rm
GL}_{2}({\mathbb{Z}})$, we can assume that $T=\begin{pmatrix}n&r/2\\\
r/2&m\end{pmatrix}$ is such that $m=min~{}T$, where $min~{}T$ is the least
positive integer represented by $T$. Now observe that $a(T)$ is the $(n,r)$-th
Fourier coefficient of $\phi_{m}$. Using Proposition 5.2.3, estimate (37) and
$m=min~{}T~{}\ll~{}({\rm det}~{}T)^{1/2}$ we complete the proof. ∎

######  Remark 5.2.5.

We would like to mention that if one could prove (37) for $F\in
S_{k}({\Gamma}_{0}^{(2)}(N))$ which is not of classical Saito–Kurokawa type,
that would lead to an improvement of the current best known unconditional
result [Koh92].

## References

  * [AB15] Mahesh Agarwal and Jim Brown.  Saito-Kurokawa lifts of square-free level.  Kyoto J. Math., 55(3):641–662, 2015.
  * [And74] A. N. Andrianov.  Euler products that correspond to Siegel’s modular forms of genus $2$.  Uspehi Mat. Nauk, 29(3 (177)):43–110, 1974.
  * [And09] Anatoli Andrianov.  Introduction to Siegel modular forms and Dirichlet series.  Universitext. Springer, New York, 2009.
  * [Art13] James Arthur.  The endoscopic classification of representations, volume 61 of American Mathematical Society Colloquium Publications.  American Mathematical Society, Providence, RI, 2013.  Orthogonal and symplectic groups.
  * [AZ95] A. N. Andrianov and V. G. Zhuravlëv.  Modular forms and Hecke operators, volume 145 of Translations of Mathematical Monographs.  American Mathematical Society, Providence, RI, 1995.  Translated from the 1990 Russian original by Neal Koblitz.
  * [BH17] C. J. Bushnell and G. Henniart.  Strong exponent bounds for the local Rankin-Selberg convolution.  Bull. Iranian Math. Soc., 43(4):143–167, 2017.
  * [Bor76] Armand Borel.  Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup.  Invent. Math., 35:233–259, 1976.
  * [BR88] S. Böcherer and S. Raghavan.  On Fourier coefficients of Siegel modular forms.  J. Reine Angew. Math., 384:80–101, 1988.
  * [Bre99] Stefan Breulmann.  On Hecke eigenforms in the Maaß space.  Math. Z., 232(3):527–530, 1999.
  * [Bri04] K. Bringmann.  Applications of Poincaré series on Jacobi groups.  PhD Thesis, University of Heidelberg, 2004.
  * [Bro10] Jim Brown.  The first negative Hecke eigenvalue of genus 2 Siegel cuspforms with level $N\geq 1$.  Int. J. Number Theory, 6(4):857–867, 2010.
  * [CGK15] YoungJu Choie, Sanoli Gun, and Winfried Kohnen.  An explicit bound for the first sign change of the Fourier coefficients of a Siegel cusp form.  Int. Math. Res. Not. IMRN, (12):3782–3792, 2015.
  * [CK21] Peter J. Cho and Henry H. Kim.  The smallest prime in a conjugacy class and the first sign change for automorphic $L$-functions.  Proc. Amer. Math. Soc., 149(3):923–933, 2021.
  * [Cor19] Andrew Corbett.  An explicit conductor formula for ${\rm GL}_{n}\times{\rm GL}_{1}$.  Rocky Mountain J. Math., 49(4):1093–1110, 2019.
  * [Das13] Soumya Das.  On the natural densities of eigenvalues of a Siegel cusp form of degree 2.  Int. J. Number Theory, 9(1):9–15, 2013.
  * [DK13] Soumya Das and Winfried Kohnen.  Some remarks on the Resnikoff-Saldaña conjecture.  In The legacy of Srinivasa Ramanujan, volume 20 of Ramanujan Math. Soc. Lect. Notes Ser., pages 153–161. Ramanujan Math. Soc., Mysore, 2013.
  * [DK18] Soumya Das and Winfried Kohnen.  On sign changes of eigenvalues of Siegel cusp forms of genus 2 in prime powers.  Acta Arith., 183(2):167–172, 2018.
  * [DKS14] Soumya Das, Winfried Kohnen, and Jyoti Sengupta.  On a convolution series attached to a Siegel Hecke cusp form of degree 2.  Ramanujan J., 33(3):367–378, 2014.
  * [DPSS20] Martin Dickson, Ameya Pitale, Abhishek Saha, and Ralf Schmidt.  Explicit refinements of Böcherer’s conjecture for Siegel modular forms of squarefree level.  J. Math. Soc. Japan, 72(1):251–301, 2020.
  * [EZ85] Martin Eichler and Don Zagier.  The theory of Jacobi forms, volume 55 of Progress in Mathematics.  Birkhäuser Boston Inc., Boston, MA, 1985.
  * [FM21] Masaaki Furusawa and Kazuki Morimoto.  Refined global Gross-Prasad conjecture on special Bessel periods and Böcherer’s conjecture.  J. Eur. Math. Soc. (JEMS), 23(4):1295–1331, 2021.
  * [Fre91] Eberhard Freitag.  Singular modular forms and theta relations, volume 1487 of Lecture Notes in Mathematics.  Springer-Verlag, Berlin, 1991.
  * [GI14] Wee Teck Gan and Atsushi Ichino.  Formal degrees and local theta correspondence.  Invent. Math., 195(3):509–672, 2014.
  * [GKP19] Sanoli Gun, Balesh Kumar, and Biplab Paul.  The first simultaneous sign change and non-vanishing of Hecke eigenvalues of newforms.  J. Number Theory, 200:161–184, 2019.
  * [GKP21] Sanoli Gun, Winfried Kohnen, and Biplab Paul.  Arithmetic behaviour of Hecke eigenvalues of Siegel cusp forms of degree two.  Ramanujan J., 54(1):43–62, 2021.
  * [GMP21] Sanoli Gun, V. Kumar Murty, and Biplab Paul.  Distinguishing newforms by their Hecke eigenvalues.  Res. Number Theory, 7(3):Paper No. 49, 24, 2021.
  * [GPS18] Sanoli Gun, Biplab Paul, and Jyoti Sengupta.  On Hecke eigenvalues of Siegel modular forms in the Maass space.  Forum Math., 30(3):775–783, 2018.
  * [GS17] S. Gun and J. Sengupta.  Sign changes of Fourier coefficients of Siegel cusp forms of degree two on Hecke congruence subgroups.  Int. J. Number Theory, 13(10):2597–2625, 2017.
  * [GT11] Wee Teck Gan and Shuichiro Takeda.  The local Langlands conjecture for ${\rm GSp}(4)$.  Ann. of Math. (2), 173(3):1841–1882, 2011.
  * [Ibu12] Tomoyoshi Ibukiyama.  Saito-Kurokawa liftings of level $N$ and practical construction of Jacobi forms.  Kyoto J. Math., 52(1):141–178, 2012.
  * [IK04] Henryk Iwaniec and Emmanuel Kowalski.  Analytic number theory, volume 53 of American Mathematical Society Colloquium Publications.  American Mathematical Society, Providence, RI, 2004.
  * [IK22] Tamotsu Ikeda and Hidenori Katsurada.  Estimates for the fourier coefficients of the duke-imamoglu-ikeda lift, 2022.
  * [Jes08] S. Jesgarz.  Vorzeichenwechsel von fourierkoeffizienten von siegelschen spitzenformen.  Diploma Thesis (unpublished), University of Heidelberg, 2008.
  * [Jin19] Seokho Jin.  On some first sign change problems of modular forms.  J. Math. Anal. Appl., 472(1):577–583, 2019.