# Radiative and meson decays of $Y(4230)$ in flavor SU(3)

Luciano Maiani  Antonio D. Polosa  Verónica Riquer  Dipartimento di Fisica and
INFN, Sapienza Università di Roma
Piazzale Aldo Moro 2, I-00185 Roma, Italy

###### Abstract

The charmonium-like exotic states $Y(4230)$ and the less known $Y(4320)$,
produced in $e^{+}e^{-}$ collisions, are sources of positive parity exotic
hadrons in association with photons or pseudoscalar mesons. We analyze the
radiative and pion decay channels in the compact tetraquark scheme, with a
method that proves to work equally well in the most studied
$D^{*}\to\gamma/\pi+D$ decays. The decay of the vector $Y$ into a pion and a
$Z_{c}$ state requires a flip of charge conjugation and isospin that is
described appropriately in the formalism used. Rates however are found to
depend on the fifth power of pion momentum which would make the final states
$\pi Z_{c}(4020)$ strongly suppressed with respect to $\pi Z_{c}(3900)$. The
agreement with BES III data would be improved considering the $\pi
Z_{c}(4020)$ events to be fed by the tail of the $Y(4320)$ resonance under the
$Y(4230)$. These results should renovate the interest in further clarifying
the emerging experimental picture in this mass region.

###### pacs:

14.40.Rt, 12.39.-x, 12.40.-y

##  I Introduction

The study of final states in $e^{+}e^{-}$ high energy annihilation, with the
pioneering contributions by BaBar, Belle and BES collaborations, has opened
the way to the new spectroscopy of exotic hadrons.

The so-called $Y$ states, unexpected charmonium-like states created by the
initial lepton pair, are efficient sources of positive parity exotic hadrons
produced in association with one photon, pion or K meson.

Decays of the lightest $Y$ states, such as $Y(4230)$ into $\gamma/\pi/K+X/Z$,
extensively studied by the BES III Collaboration, have provided precious
informations on properties and quantum numbers of the lightest, $J^{P}=1^{+}$
exotic states (see e.g. changzheng ), the latest result being the observation
of the first, hidden charm, open strangeness $Z_{cs}(3985)$, produced in
association with a charged K meson in Ablikim:2020hsk  (exotic hadrons are
extensively reviewed in Ali:2019roi ; Chen:2016qju ; Esposito:2016noz ;
Ali:2017jda ; Guo:2017jvc ; Lebed:2016hpi ; Olsen:2017bmm ).

In this note, we adopt the compact tetraquark model for
$X(3872),~{}Z_{c}(3900),~{}Z_{c}(4020)$, as $S$-wave tetraquarks Maiani:2004vq
; Maiani:2014aja ; Maiani:2016wlq , and for $Y$ states, as $P$-wave
tetraquarks Maiani:2014aja ; Ali:2017wsf , to study radiative and pionic
decays of $Y(4230)$

$\displaystyle Y(4230)\to\gamma+X(3872)$ (1) $\displaystyle
Y(4230)\to\pi+Z_{c}(3900)/Z_{c}(4020),$ (2)

observed by BES III in the reactions

$e^{+}e^{-}\to Y(4230)\to\pi+Z_{c}(3900)/Z_{c}(4020)~{}{\rm
or}~{}\gamma+X(3872)$ (3)

For a $Y$ resonance of valence composition $[cu][\bar{c}\bar{u}]$ or
$[cd][\bar{c}\bar{d}]$, the photon in (1) is emitted from the light quark or
antiquark. Decay (2) arises from the elementary transitions

$u\to d~{}\pi^{+}\quad{\rm or}\quad\bar{d}\to\bar{u}~{}\pi^{+}$ (4)

and similar for $\pi^{-}$ and $\pi^{0}$. The same transitions are operative in
$D^{*}\to\gamma/\pi+D$ decays Casalbuoni:1996pg .

Our results for $Z_{c}(3900)$ and $X(3872)$ are in quantitative agreement with
earlier studies of $D^{*}$ decays. The agreement is, of course, welcome but
not unexpected and it supports the picture of compact tetraquarks bound by QCD
forces.

We find a strong dependence of decay rates from the pion momentum,
$\Gamma\propto q^{5}$. As a consequence, the decay $Y(4230)\to\pi Z_{c}(4020)$
is strongly suppressed with respect to the decay into $\pi Z_{c}(3900)$, which
does not seem to be supported by the cross sections reported by BES III. One
possible explanation could be that the $Z_{c}(4020)$ events come from the
second peak, $Y(4320)$. A clarification of the distribution of $\pi
D^{*}\bar{D}^{*}$ events in the region as well as information on the decay
modes of $Y(4320)$ would be very useful.

Production of exotic states in $e^{+}e^{-}$ annihilation goes essentially via
$Y$ resonances. It is reasonable to assume that the open strangeness state
$Z_{cs}(3985)$ seen in Ablikim:2020hsk

$e^{+}e^{-}\to K^{+}Z^{-}_{cs}(3985)\to
K^{+}(D_{s}^{*-}D^{0}+D_{s}^{-}D^{*0})$ (5)

also arises from a $Y$-like resonance with $[cu][\bar{c}\bar{u}]$ or
$[cs][\bar{c}\bar{s}]$ valence quark composition that decays to the final
$K^{+}Z_{cs}$ state by the elementary processes

$u\to s~{}K^{+}\quad{\rm or}\quad\bar{s}\to\bar{u}~{}K^{+}$ (6)

If the hypothesis is correct, our analysis of K meson transitions, shows that
strange members of the two nonets associated to $X(3872)$ and $Z_{c}(3900)$
should both appear in the final states of (5), i.e. the
$D_{s}\bar{D}^{*0}+c.c.$ spectrum should include as well the $Z_{cs}(4003)$
recently observed by LHCb in $B^{+}$ decay Aaij:2021ivw  (the classification
of the newly discovered $Z_{cs}$ resonances is considered in Maiani:2021tri ).
This is a crucial feature that can be tested in higher luminosity experiments.

##  II Production and decay modes of $Y(4230)$ in $e^{+}e^{-}$ annihilation

A $J^{PC}=1^{--}$ resonance, $Y(4620)$, was first observed by BaBar and
confirmed by Belle in $e^{+}e^{-}$ annihilation with Initial State Radiation
(ISR)Aubert:2005rm ; Wang:2007ea . BES III has later studied the $4620$
structure with higher resolution and shown that it is resolved in two lines,
now indicated as $Y(4230)$ and $Y(4320)$ (see changzheng ).

$Y$ states as $P$-wave tetraquarks have been described in Maiani:2014aja ;
Ali:2017wsf . One expects four states $Y_{1},\dots,Y_{4}$, the two lightest
ones with spin composition

$\displaystyle Y_{1}=|(0,0),L=1\rangle_{J=1}$ (7) $\displaystyle
Y_{2}=\frac{1}{\sqrt{2}}\big{(}|(1,0),L=1\rangle+|(0,1),L=1\rangle\big{)}_{J=1}$
(8)

Valence quark composition is $[cq][\bar{c}\bar{q}]$, diquark and antidiquark
spin are indicated in parenthesis, $L$ is the orbital angular momentum.

It was noted in Ali:2017wsf  that the mass difference of $Y_{1,2}$ arises from
two contrasting contributions: the hyperfine interaction, which pushes $Y_{1}$
down, and the spin-orbit interaction, which pushes $Y_{2}$ down. We had chosen
$M_{1}<M_{2}$ on the basis of a preliminary indication that the
$\gamma+X(3872)$ decay was associated to $Y(4320)$, since this decay may arise
from the $Y_{2}$ structure in Eq. (8) and not from $Y_{1}$, Eq. (7).

Later information Ablikim:2019zio  indicates that the source of the
$\gamma+X(3872)$ decay is instead $Y(4230)$. Consequently, we are led to
change the assignment and propose $M_{2}<M_{1}$, that is

$Y_{2}=Y(4230)\quad Y_{1}=Y(4320)~{}{\rm or~{}higher}$ (9)

Ref. |  $Z$(Mass) |  $\sqrt{s}$ (GeV) | $e+e\to Y(4230)\to\dots$ |  $Q$-value |  $\sigma$ (pb)
---|---|---|---|---|---
PRL 115Ablikim:2015gda  | $Z_{c}(3885)$ | $4.226$ | $\pi^{0}Z_{c}^{0}\to\pi^{0}(D{\bar{D}}^{*}+c.c.)^{0}$ | $197$ | $77\pm 21$
PRD 92Ablikim:2015swa  | $Z_{c}(3885)$ | $4.23$ | $[\pi^{+}Z_{c}^{-}+c.c.]\to[\pi^{+}(D\bar{D}^{*})^{-}+c.c.]$ | $197$ | $141\pm 14$
PRL 112 Ablikim:2013emm  | $Z_{c}(4020)$ | $4.26$ | $[\pi^{+}Z_{c}^{-}+c.c.]\to[\pi^{+}(D^{*}\bar{D}^{*})^{-}+c.c.]$ | $65$ | $\begin{array}[]{c}(0.65\pm 0.11)\cdot(137\pm 17)=\\\ =89\pm 19\end{array}$
PRL 115 Ablikim:2015vvn  | $Z_{c}(4020)$ | $4.23$ | $\pi^{0}Z_{c}^{0}\to\pi^{0}(D^{*}\bar{D}^{*})^{0}$ | $65$ | $62\pm 12$
PRL 115Ablikim:2013dyn  | $X(3872)$ | $4.226$ | $\gamma X\to\gamma\pi^{+}\pi^{-}J/\psi$ | $--$ | $0.27\pm 0.09\pm 0.12$
PRD100Li:2019kpj  |  |  | $\gamma X$ | $354$ | $5.5^{+2.8}_{-3.6}$
2011.07855Ablikim:2020hsk  | $Z_{cs}(3982)$ | $4.681$ | $K^{+}Z_{cs}^{-}\to K^{+}(D^{*-}_{s}D^{0}+D^{-}_{s}D^{*0})$ | $199$ | $4.4\pm 0.9$
Table 1: $e^{+}e^{-}$ annhilation cross sections into exotic hadrons,
determined by BES III in the $Y(4230)$ region.

Table 1 summarizes the cross sections of different final states produced in
$e^{+}e^{-}$ annihilation at the $Y(4230)$ peak. Cross sections are related to
the width $\Gamma(Y(4230)\to f)$ by the formula

$\sigma_{\rm peak}(e^{+}e^{-}\to Y(4230)\to
f)=\frac{12\pi}{M_{Y}^{2}}\frac{\Gamma_{e}\Gamma_{f}}{\Gamma^{2}}$ (10)

($\Gamma_{e}$ is the width to an electron pair). The total width of $Y(4230)$
is estimated in changzheng

$\Gamma(Y(4230))=(56.0\pm 3.6\pm 6.9)~{}{\rm MeV}$ (11)

Data from BES III indicate that $Y(4230)$ is isoscalar BESIII:2020pov ;
Collaboration:2017njt . Thus, denoting by $Y_{u,d}$ the $Y_{2}$ states with
$u\bar{u}$ and $d\bar{d}$ valence quarks, we take

$Y(4230)=Y_{2}=\frac{Y_{u}+Y_{d}}{\sqrt{2}}$ (12)

##  III $Y(4230)$ transitions to $S$-wave tetraquarks

We consider the decays

$\displaystyle Y_{2}\to\gamma+X$ (13) $\displaystyle Y_{2}\to\pi+Z$ (14)
$\displaystyle Y_{2}\to\pi+Z^{\prime}$ (15)

where $X,~{}Z$ and $Z^{\prime}$ are the $S$-wave tetraquarks

$\displaystyle X$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\big{(}|(1,0),~{}L=0\rangle+|(0,1),~{}L=0\rangle\big{)}_{J=1}$
(16) $\displaystyle Z$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\big{(}|(1,0),~{}L=0\rangle-|(0,1),~{}L=0\rangle\big{)}_{J=1}$
(17) $\displaystyle Z^{\prime}$ $\displaystyle=$
$\displaystyle|(1,1),~{}L=0\rangle_{J=1}$ (18)

The decay (13) as a dipole transition $L=1\to L=0,~{}\Delta S=0$ has been
considered in Chen:2015dig . Here we rederive the result as an introduction to
pionic transitions.

We work in the non-relativistic approximation and describe the states with
wave functions in spin and coordinate space. In the rest frame of $Y_{2}$

$|Y_{2}^{a}\rangle=N_{Y}\Big{[}\epsilon_{abc}S^{b(+)}\frac{\xi^{c}}{r}R_{2P}(r)\Big{]}$
(19)

$N_{Y}$ is a normalization constant, the spin wave function from (8) and (16)
is

$S^{a(+)}=\frac{(c\sigma_{2}\sigma^{a}u)_{\bm{x}}(\bar{c}\sigma_{2}\bar{u})_{\bm{y}}+(c\sigma_{2}u)_{\bm{x}}(\bar{c}\sigma_{2}\sigma^{a}\bar{u})_{\bm{y}}}{2\sqrt{2}}$
(20)

We indicate with a bar the charge-conjugate quark fields, ${\bm{x,y}}$ are
diquark and antidiquark coordinates, ${\bm{\xi=x-y}}$ the relative coordinate
and $r=|{\bm{\xi}}|$ the relative radius. The plus sign in $S^{a(+)}$ reminds
of the charge conjugation, as defined on the rhs of (20).

Considering the decay into $X$, we take

$|X^{a}\rangle=N_{X}~{}\Big{[}S^{a(+)}~{}R_{1S}(r)\Big{]}$ (21)

and normalize spin w.f. according to

$S^{a(+)}\cdot S^{b(+)}=\delta_{ab}$ (22)

Thus

$\displaystyle\delta_{ab}$ $\displaystyle=$ $\displaystyle\langle
Y_{2}^{a}|Y_{2}^{b}\rangle=N_{Y}^{2}~{}\delta_{ab}\frac{8\pi}{3}\int
dr~{}y_{2P}(r)^{2}$ $\displaystyle\delta_{ab}$ $\displaystyle=$
$\displaystyle\langle X^{a}|X^{b}\rangle=N_{X}^{2}~{}\delta_{ab}4\pi\int
dr~{}y_{1S}(r)^{2}$

with $y(r)=rR(r)$, and $R(r)$ the radial wave function.

_Radiative decay._ We work in the radiation gauge, $A^{0}=0$ and
${\bm{\nabla}}\cdot{\bm{A}}=0$. The photon couples to $u$ and to other quarks
with the basic lagrangian

${\cal L}_{\rm e.m.}=eQ_{u}\,\bar{u}({\bm{x}})\,{\bm{\gamma\cdot
A}}({\bm{x}})\,u({\bm{x}})+(u\to d)$ (23)

The elementary transition amplitudes are

$\displaystyle{\cal M}_{u}$ $\displaystyle=$ $\displaystyle
eQ_{u}\,\chi^{\dagger}\left[\frac{({\bm{p}+\bm{q}})\cdot{\bm{\sigma}}}{2m_{u}}\sigma^{a}+\sigma^{a}\frac{{\bm{p}}\cdot{\bm{\sigma}}}{2m_{u}}\right]\chi\,\epsilon^{a}=eQ_{u}\,\chi^{\dagger}\left[\frac{{\bm{p}}_{u}{\bm{\cdot\epsilon}}}{m_{u}}+\frac{i{\bm{q\wedge\epsilon\cdot\sigma}_{u}}}{2m_{u}}\right]\chi$
(24) $\displaystyle{\cal M}_{\bar{u}}$ $\displaystyle=$ $\displaystyle-
eQ_{u}\,\chi^{\dagger}\left[\frac{({\bm{p}+\bm{q}})\cdot{\bm{\sigma}}}{2m_{u}}\sigma^{a}+\sigma^{a}\frac{{\bm{p}}\cdot{\bm{\sigma}}}{2m_{u}}\right]\chi\,\epsilon^{a}=-eQ_{u}\,\chi^{\dagger}\left[\frac{{\bm{p}}_{\bar{u}}{\bm{\cdot\epsilon}}}{m_{\bar{u}}}+\frac{i{\bm{q\wedge\epsilon\cdot\sigma}_{\bar{u}}}}{2m_{u}}\right]\chi$
(25)

The minus sign in ${\cal M}_{\bar{u}}$ arises from charge conjugation. In view
of large mass denominators, we neglect radiation from the charm quarks.

The right-hand sides of these equations contain products of operators acting
on the spin and space wave functions of the initial tetraquark multiplied by
variables of the electromagnetic field. As usual, we identify

${\bm{p}}_{u}=-i{\bm{\partial}}_{\bm{x}}\qquad{\bm{p}}_{\bar{u}}=-i{\bm{\partial}}_{\bm{y}}$
(26)

Acting on functions of ${\bm{\xi}=\bm{x-y}}$

${\bm{\partial}}_{\bm{x}}={\bm{\partial}}_{\bm{\xi}}=i{\bm{p}}_{\bm{\xi}}\qquad{\bm{\partial}}_{\bm{y}}=-{\bm{\partial}}_{\bm{\xi}}=-i{\bm{p}}_{\bm{\xi}}$
(27)

Further, we set

$\frac{{\bm{p}}_{\bm{\xi}}}{m_{u}}={\bm{v}}=\frac{d{\bm{\xi}}}{dt}=i\omega{\bm{\xi}}\qquad\omega=|{\bm{q}}|$
(28)

and the hamiltonian acting on tetraquark wave functions is

$\displaystyle H_{I}=eQ_{u}~{}\Big{[}{\bm{\xi\cdot
E}}+\frac{1}{2m_{u}}\Big{(}{\bm{\sigma}_{u}}-{\bm{\sigma}_{\bar{u}}}\Big{)}\cdot{\bm{B}}\Big{]}$
(29)

with ${\bm{E}}$ and ${\bm{B}}$ the electric and magnetic fields. The first
term corresponds to the well known electric dipole transition that changes by
one unit the orbital angular momentum, leaving the spin wave function
unchanged landau4 . One obtains

$\displaystyle{\cal M}^{ab}(Y_{2}\to\gamma+X)$ $\displaystyle=$ $\displaystyle
eQ_{u}~{}\langle
X^{a}|\xi^{c}|Y_{2}^{b}\rangle~{}i\omega\epsilon^{c}=\epsilon_{abc}~{}\epsilon^{c}~{}\frac{\omega}{\sqrt{6}}~{}\langle
r\rangle_{2P\to 1S}$ $\displaystyle\langle r\rangle_{2P\to 1S}$
$\displaystyle=$ $\displaystyle\frac{\int
dr~{}[y_{1S}(r)~{}r~{}y_{2P}(r)]}{\sqrt{\int dr~{}y_{1S}(r)^{2}\int
dr~{}y_{2P}(r)^{2}}}$ (30)

with $\omega=M_{Y}-M_{X}=\omega_{X}$ and

$\displaystyle\Gamma(Y_{2}\to\gamma+X)=\frac{4\alpha}{9}~{}Q_{u}^{2}~{}\omega_{X}^{3}\langle
r\rangle_{2P\to 1S}^{2}$ (31)

For isoscalar $Y(4230)$ we use (12). Summing incoherently over the final
states $X_{u}$ and $X_{d}$, see Maiani:2017kyi , we get

$Q_{u}^{2}\to Q_{eff}^{2}=\frac{1}{2}(Q_{u}^{2}+Q_{d}^{2})=\frac{5}{18}$

and

$\displaystyle\Gamma(Y(4320)\to\gamma+X)=0.322~{}{\rm
MeV}~{}(\omega_{X}~{}\langle r\rangle_{2P\to 1S})^{2}$ (32)

$\pi^{0}$ _emission._ We assume that quarks couple to pions via the isovector,
axial vector current 111To our knowledge, the quark-pion, axial vector
interaction to describe pionic hadron decays has been first introduced in
weisskopf .:

$\displaystyle{\cal
L}^{\pi}_{I}=~{}\frac{g}{f_{\pi}}~{}\bar{q}\gamma^{\mu}\gamma_{5}(\partial_{\mu}{\bm{\pi}})q$
(33)
$\displaystyle{\bm{\pi}}=\frac{{\tau^{i}\pi^{i}}}{\sqrt{2}}=\begin{pmatrix}\frac{\pi^{0}}{\sqrt{2}}&\pi^{+}\\\
\pi^{-}&-\frac{\pi^{0}}{\sqrt{2}}\end{pmatrix}$ (34)

We follow Casalbuoni:1996pg  for the definition of the coupling $g$ and

$f_{\pi}=132~{}{\rm MeV}$ (35)

The lagrangian contains the time derivative of the pion field. Applying the
Legendre transformation, the interaction hamiltonian is

${\cal H}^{\pi}_{I}=~{}\frac{g}{f_{\pi}}~{}{\bm{A\cdot\nabla\pi}}$ (36)

The elementary quark transition is determined by

$\displaystyle{\bm{q\cdot}}{\bm{A}}_{u}=\bar{u}(p+q)\,{\bm{q\cdot\gamma}}\gamma_{5}\,u(p)=[{\bm{q\cdot\sigma}}+\frac{q^{2}}{4m_{u}}\frac{\bm{p}}{m_{u}}\cdot{\bm{\sigma}}+\dots]$
(37)

The first term corresponds to $\Delta L=0$, operative in $D^{*}\to D\pi$
Casalbuoni:1996pg , the second to $\Delta L=1$, for $Y$ and $D_{1}$ pionic
decay, dots indicate terms with $\Delta L>1$. Using charge conjugation
symmetry, restricting to the $\Delta L=1$ term and specializing to the
$\pi^{0}$ case, we obtain the hamitonian

$H^{\pi}_{I}=\frac{g}{f_{\pi}}\frac{q^{2}}{4m_{u}}\,i\omega\,\chi^{\dagger}[{\bm{\xi\cdot}}({\bm{\sigma}}_{u}-{\bm{\sigma}}_{\bar{u}})-(u\to
d)]\chi\,\frac{\pi^{0}}{\sqrt{2}}$ (38)

We note the results of applying the spin operators to the components of the
spin wave function

$\displaystyle{\sigma}^{b}_{u}(c\sigma_{2}u)$ $\displaystyle=$
$\displaystyle(c\sigma_{2}{\sigma}^{b}u)$
$\displaystyle{\sigma}^{b}_{u}(c\sigma_{2}\sigma^{a}u)$ $\displaystyle=$
$\displaystyle(c\sigma_{2}\sigma^{a}{\sigma}^{b}u)=\Big{[}\delta_{ab}(\bar{c}\sigma_{2}\bar{u})+i\epsilon_{abc}(\bar{c}\sigma_{2}\sigma^{c}\bar{u})\Big{]}$
$\displaystyle-\sigma_{\bar{u}}^{b}(\bar{c}\sigma_{2}\bar{u})$
$\displaystyle=$ $\displaystyle-(\bar{c}\sigma_{2}\sigma^{b}{\bar{u}})$
$\displaystyle-\sigma_{\bar{u}}^{b}(\bar{c}\sigma_{2}\sigma^{a}\bar{u})$
$\displaystyle=$
$\displaystyle-(\bar{c}\sigma_{2}\sigma^{a}{\sigma}^{b}\bar{u})=-\Big{[}\delta_{ab}(\bar{c}\sigma_{2}\bar{u})+i\epsilon_{abc}(\bar{c}\sigma_{2}\sigma^{c}\bar{u})\Big{]}$
(39)

Defining

$\displaystyle S_{u}^{a(-)}$ $\displaystyle=$
$\displaystyle\frac{(c\sigma_{2}\sigma^{a}u)(\bar{a}\sigma_{2}\bar{u})-(c\sigma_{2}u)(\bar{c}\sigma_{2}\sigma^{a}\bar{u})}{2\sqrt{2}}=Z$
(40) $\displaystyle S_{u}^{a}$ $\displaystyle=$ $\displaystyle
i\epsilon_{abc}\frac{(c\sigma_{2}\sigma^{b}u)(\bar{c}\sigma_{2}\sigma^{c}\bar{u})}{2\sqrt{2}}=Z^{\prime}$
(41)

we obtain

$\displaystyle(\sigma_{u}^{b}-\sigma_{\bar{u}}^{b})S_{u}^{a(+)}=i\epsilon_{abc}\Big{[}S_{u}^{c(-)}+S_{u}^{c}\Big{]}$
(42)

Note that, going from $Y$ to $Z$ or $Z^{\prime}$, the minus sign between
$\sigma_{u}$ and $\sigma_{\bar{u}}$ changes the charge conjugation sign of the
spin w.f.. Similarly, the minus sign between the $u$ and $d$ term in (38)
changes the $S_{u}$ and $S_{d}$ combination from $I=0$ (in $Y$) to $I=1$ (in
$Z$ and $Z^{\prime}$).

In conclusion, we find

${\cal M}^{ab}=\langle
Z^{a}|H^{\pi}_{I}|Y_{2}^{b}\rangle=\delta_{ab}\sqrt{\frac{2}{3}}\frac{g}{\sqrt{2}}~{}\left(\frac{q^{2}}{4f_{\pi}m_{u}}\right)~{}(\omega_{Z}\langle
r\rangle_{2P\to 1S})$ (43)

and

$\displaystyle\Gamma(Y(4230)\to Z^{0}\pi^{0})$ $\displaystyle=$
$\displaystyle|{\cal M}^{11}|^{2}\frac{1}{(2\pi)^{3}}~{}4\pi\int\frac{q\omega
d\omega}{2\omega}~{}(2\pi)\delta(\Delta M-\omega)=\frac{q}{2\pi}~{}|{\cal
M}^{11}|^{2}=$ (44) $\displaystyle=$
$\displaystyle\frac{q}{6\pi}g^{2}~{}\left(\frac{q^{2}}{4f_{\pi}m_{u}}\right)^{2}\left(\frac{\omega_{Z}}{\omega_{X}}\right)^{2}~{}(\omega_{X}\langle
r\rangle_{2P\to 1S})^{2}=4.36~{}{\rm MeV}~{}g^{2}(\omega_{X}\langle
r\rangle_{2P\to 1S})^{2}$

$q$ is the decay momentum, $\omega_{Z}=M_{Y}-M_{Z}$, we have chosen to
normalize the radius with $\omega_{X}$, for comparison with Eq. (32) and
$m_{u}=308$ MeV, from the constituent quark model spectrum of mesons (see e.g.
Maiani:2004vq ; Ali:2019roi ).

##  IV Charge Conjugation in $Y$ and other tetraquark nonets

A charge conjugation quantum number can be given to each self conjugate SU(3)f
multiplet according to

${\cal C}T{\cal C}=\eta_{T}{\tilde{T}}$ (45)

where ${\cal C}$ denotes the operator of charge conjugation, $T$ the matrix
representing the multiplet in SU(3) space and ${\tilde{T}}$ the transpose
matrix. $\eta_{T}$ is the sign taken by neutral members, but it can be
attributed to all members of the multiplet. In the exact SU(3)f limit, $\eta$
is conserved in strong and electromagnetic decays. $\eta=-1$ is given to the
electromagnetic current $J^{\mu}$ and to $Y^{\mu}$ while $\eta_{K,\pi}=+1$.

We extend $Y_{2}$ to a full nonet that we write as (omitting the overall
normalization for brevity)

$\displaystyle Y_{2}(a,\bar{b})^{\gamma}$ $\displaystyle=$
$\displaystyle\epsilon_{\gamma\alpha\beta}~{}\Big{[}(D_{a}^{\alpha}\bar{D}_{\bar{b}})+(D_{a}\bar{D}_{\bar{b}}^{\alpha})\Big{]}~{}\frac{\xi^{\beta}}{r}~{}F(\xi)$
$\displaystyle D_{a}^{\alpha}$ $\displaystyle=$
$\displaystyle(c\sigma_{2}\sigma^{\alpha}q_{a});~{}\bar{D}_{\bar{b}}=(\bar{c}\sigma_{2}\bar{q}_{b}),~{}{\rm
etc.}$ (46)

$F(\xi)$ is the wave function in the relative coordinate, even under
$\xi\to-\xi$.

The lagrangian (33) generalizes to

${\cal
L}^{\pi}_{I}=~{}\frac{g}{f_{\pi}}~{}\bar{q}\gamma^{\mu}\gamma_{5}(\partial_{\mu}{\bm{M}})q$
(47)

where

${\bm{M}}=\frac{{\lambda}^{i}M^{i}}{\sqrt{2}}=\begin{pmatrix}\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta_{8}}{\sqrt{6}}&\pi^{+}&{\footnotesize{K^{+}}}\\\
\pi^{-}&-\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta_{8}}{\sqrt{6}}&{\footnotesize{K^{0}}}\\\
{\footnotesize{K^{-}}}&{\footnotesize{\bar{K}^{0}}}&{\footnotesize{-2}}\frac{\eta_{8}}{\sqrt{6}}\end{pmatrix}$
(48)

($\lambda^{i}$ are the Gell-Mann matrices). Correspondingly, the action on
each diquark of the hamiltonian derived from (47) is (for brevity, we omit
two-dimensional spinors $\chi^{\dagger}$ and $\chi$, which should bracket all
the expressions below)

$\displaystyle{\cal H}_{I}D_{a}^{\alpha}$ $\displaystyle\propto$
$\displaystyle
M_{aa^{\prime}}({\bm{\sigma\cdot\xi}})^{\alpha\beta}D_{a^{\prime}}^{\beta}=(MD_{\sigma\xi})_{a}^{\alpha}$
$\displaystyle{\cal H}_{I}\bar{D}_{\bar{b}}$ $\displaystyle\propto$
$\displaystyle-({\bm{\sigma\cdot\xi}})(M_{\bar{b}^{\prime}\bar{b}}\bar{D})_{\bar{b}^{\prime}}=-({\bm{\sigma\cdot\xi}})\bar{D}_{\bar{b}^{\prime}}M_{\bar{b}^{\prime}\bar{b}}=-(\bar{D}_{\sigma\xi}M)_{\bar{b}},~{}{\rm
etc.}$

and

${\cal
H}_{I}(D_{a}^{\alpha}\bar{D}_{\bar{b}})\propto(MD_{\sigma\xi})^{\alpha}_{a}\bar{D}_{\bar{b}}+D_{a}^{\alpha}(\bar{D}_{\sigma\xi}M)_{\bar{b}}$
(49)

where

$(D_{\sigma\xi})^{\alpha}_{a}=(c\sigma_{2}(\sigma\cdot\xi)q_{a})\qquad(\bar{D}_{\sigma\xi})_{\bar{b}}=(\bar{c}\sigma_{2}(\sigma\cdot\xi)\bar{q}_{\bar{b}})$

Explicitly,

$\displaystyle\frac{\xi^{\beta}}{r}~{}\Big{[}(D_{\sigma\xi})^{\alpha}_{a}\bar{D}_{\bar{b}}+(D_{\sigma\xi})_{a}\bar{D}^{\alpha}_{\bar{b}}\Big{]}=\frac{\xi^{\beta}}{r}\xi^{\rho}~{}\Big{[}(c\sigma_{2}\sigma^{\alpha}\sigma^{\rho}q_{a})(\bar{c}\sigma_{2}\bar{q}_{\bar{b}})+(c\sigma_{2}\sigma^{\rho}q_{a})(\bar{c}\sigma_{2}\sigma^{\alpha}q_{\bar{b}})\Big{]}=$
$\displaystyle=\frac{\xi^{\beta}}{r}\xi^{\rho}~{}\Big{[}\delta^{\alpha\rho}~{}(c\sigma_{2}q_{a})(\bar{c}\sigma_{2}\bar{q}_{\bar{b}})+i\epsilon^{\alpha\rho\nu}(c\sigma_{2}\sigma^{\nu}q_{a})(\bar{c}\sigma_{2}\bar{q}_{\bar{b}})+(c\sigma_{2}\sigma^{\rho}q_{a})(\bar{c}\sigma_{2}\sigma^{\alpha}q_{\bar{b}})\Big{]}$
(50)

Applying similar arguments to the second line of (IV), we obtain

$\displaystyle\frac{\xi^{\beta}}{r}~{}\Big{[}(D)^{\alpha}_{a}(\bar{D}_{\sigma\xi})_{\bar{b}}+(D)_{a}(\bar{D}_{\sigma\xi})^{\alpha}_{\bar{b}}\Big{]}=\frac{\xi^{\beta}}{r}\xi^{\rho}~{}\Big{[}(c\sigma_{2}\sigma^{\alpha}q_{a})(\bar{c}\sigma_{2}\sigma^{\rho}\bar{q}_{\bar{b}})+(c\sigma_{2}q_{a})(\bar{c}\sigma_{2}\sigma^{\alpha}\sigma^{\rho}q_{\bar{b}})\Big{]}=$
$\displaystyle=\frac{\xi^{\beta}}{r}\xi^{\rho}~{}\Big{[}(c\sigma_{2}\sigma^{\alpha}q_{a})(\bar{c}\sigma_{2}\sigma^{\rho}\bar{q}_{\bar{b}})+\delta^{\alpha\rho}~{}(c\sigma_{2}q_{a})(\bar{c}\sigma_{2}q_{\bar{b}})+i\epsilon_{\alpha\rho\nu}(c\sigma_{2}q_{a})(\bar{c}\sigma_{2}\sigma^{\nu}q_{\bar{b}})\Big{]}$
(51)

The expressions in (50) and (51) are to be integrated with functions symmetric
under $\xi^{a}\to-\xi^{a}$, so we can replace
$\xi^{\beta}\xi^{\rho}\to\delta^{\beta\rho}~{}r^{2}/3$. In addition, in the
square brackets we can add and subtract terms that reconstruct the spin wave
functions of tetraquarks of charge conjugation $+1$, spin $0,1,0,2$, namely
$X,~{}X_{0},~{}X^{\prime}_{0},X_{2}$, and of charge conjugation $-1$, spin
$1$, i.e. $Z,~{}Z^{\prime}$. Indicating for brevity only
$X,~{}Z,~{}Z^{\prime}$, Eqs. (16) (17) and (18), we obtain

$\displaystyle{\cal
H}_{I}~{}Y_{2}\propto\frac{r}{3}~{}i\epsilon_{\alpha\beta\nu}~{}\frac{1}{2}\Big{[}(MX^{\nu}-X^{\nu}M)+(MZ^{\nu}+Z^{\nu}M)+(MZ^{\prime\nu}+Z^{\prime\nu}M)+\dots\Big{]}$
(52)

Multiplying by the $SU(3)$ matrix representing $Y$ and taking the trace we
obtain the exact SU(3)f rules for the couplings of a $C=-1$ vector nonet to
$M$ plus an $S$-wave tetraquark of charge conjugation $\eta_{T}$:

$\displaystyle{\cal H}_{I}$ $\displaystyle\propto$ $\displaystyle{\rm
Tr}[Y[M,X]]\quad(\eta_{X}=+1)$ $\displaystyle{\cal H}_{I}$
$\displaystyle\propto$ $\displaystyle{\rm Tr}[Y\\{M,Z\\}]\quad(\eta_{Z}=-1).$
(53)

In particular, for $Y(4230)$: $Y={\rm diag}(1/\sqrt{2},1/\sqrt{2},0)$, we
obtain vanishing coupling $Y\to\pi X$ and Eq. (43) for $Y_{2}\to\pi^{0}Z^{0}$.

Summarizing, we obtain the selection rules:

  1. 1.

$Y_{2}(I=0)$ does not decay into $\pi^{\pm,0}X^{\mp,0}$

  2. 2.

$Y_{2}(I=1)$ decays into $\pi^{+}X^{-}-\pi^{-}X^{+}$

  3. 3.

$Y_{2}(I=0)$ decays into $\pi^{+}Z^{-}+\pi^{-}Z^{+}+\pi^{0}Z^{0}$, same for
$Z^{\prime}$

  4. 4.

$Y_{2}(I=0,{\rm or}~{}1)$ or $Y[cs\bar{c}\bar{s}]$ all decay into
$(K^{+}X_{cs}^{-}~{}-~{}c.c.)$ and $(K^{+}Z_{cs}^{-}~{}+~{}c.c.)$

  5. 5.

The decay $Z_{cs}\to J/\psi\,K$ is allowed in the exact SU(3)f limit with

${\cal H}_{I}=\lambda\mu~{}\psi~{}({\rm Tr}\\{Z,M\\}),~{}[\mu]={\rm mass}$
(54)

  6. 6.

The decay $X_{cs}\to J/\psi\,K$ may occur to first order in SU(3)f symmetry
breaking with

${\cal H}_{I}=\lambda~{}i\psi~{}{\rm
Tr}([\epsilon_{8}[X,M])\sim\lambda~{}(m_{s}-m_{u})~{}i\psi(X_{cs}^{+}K^{-}~{}-~{}c.c.)$
(55)

##  V Radiative and pionic decays: ${{D^{*}}}$ and ${{D_{1}}}$ mesons

_The decay_ $\bm{D^{*}\to\gamma+D}$. In its spin dependent part, the
hamiltonian (29) describes the radiative decay of $D^{*}$, $\Delta S=1$ and no
change in orbital angular momentum. Setting the charm quark in the origin,
$D^{*}$ and $D$ are represented by

$\displaystyle
D^{(*a)}={\bm{V}}^{(a)}\cdot\Big{(}c^{\dagger}(0)\frac{{\bm{\sigma}}}{\sqrt{2}}u({\bm{x}})\Big{)}~{}R(|{\bm{x}}|)\qquad
D=\Big{(}c^{\dagger}(0)\frac{1}{\sqrt{2}}u({\bm{x}})\Big{)}~{}R(|{\bm{x}}|)$
$\displaystyle{\cal
M}(D^{*0}\to\gamma+D^{0})=\frac{eQ_{u}}{2m_{u}}\chi^{\dagger}\,{\bm{V}}^{(a)}\cdot{\bm{q\wedge\epsilon}^{(b)}}\,\chi$

and we obtain

$\displaystyle\Gamma(D^{*0}\to\gamma+D^{0})=\frac{\alpha}{3}~{}\left(\frac{Q_{u}}{m_{u}}\right)^{2}~{}q^{3}$
(56)

Ref. Casalbuoni:1996pg  can be consulted for a discussion of the $D^{*}$ decay
rate and the strong interaction corrections to (56).

_The decay_ ${\bm{D^{*+}\to\pi^{0}D^{+}}}$. From the hamiltonian (36) and Eq.
(37), the relevant term in the hamiltonian is

${\cal H}_{I}^{\pi}=\frac{g}{\sqrt{2}f_{\pi}}\,{\bm{q\cdot\sigma_{u}}}$

so that

${\cal M}(D^{*+}\to\pi^{0}D^{+})=\frac{g}{\sqrt{2}f_{\pi}}~{}{\bm{V\cdot q}}$

and

$\displaystyle\Gamma(D^{*+}\to\pi^{0}D^{+})=g^{2}~{}\frac{(p^{0})^{3}}{12\pi
f_{\pi}^{2}}$

with $p^{0}$ the decay momentum. Also

$\Gamma(D^{*+}\to\pi^{+}D^{0})=g^{2}\frac{(p^{+})^{3}}{6\pi f_{\pi}^{2}}$

(decay momentum $p^{+}$). We reproduce the results of Casalbuoni:1996pg . We
assume $D^{*+}$ decay to be dominated by $\pi D$ final states and use the
$D^{*+}$ total width pdg  to estimate the value of $g$, obtaining

$g\sim 0.56.$ (57)

${\bm{D^{0}_{1}\to\pi^{0}~{}D^{*0}}}$ _transition_. $D_{1}(2420)$ is a well
identified $P$-wave, positive parity charmed meson with total spin and angular
momentum $S=J=1$. We can use its decay into $D^{*}\pi$ to calibrate the
$\Delta L=1$ hamiltonian (38). In analogy with (19), we write the $D_{1}$ wave
function as:

$\displaystyle|D_{1}^{a}\rangle$ $\displaystyle=$ $\displaystyle
N_{1}\Big{[}\epsilon_{abc}\frac{\xi^{b}}{r}~{}(\bar{c}\sigma_{2}\frac{\sigma^{c}}{\sqrt{2}}u)~{}R_{2P,D}(r)\Big{]}$
(58) $\displaystyle\delta^{ab}$ $\displaystyle=$ $\displaystyle\langle
D_{1}^{a}|D_{1}^{b}\rangle=N_{1}^{2}~{}\frac{8\pi}{3}~{}\int~{}dr~{}y_{2P,D}(r)^{2}$

where the subscript $D$ indicates that the QCD couplings of the $\bar{c}u$
system are used.

The decay is induced by the ${\bm{\xi}}$ dependent part of the hamiltonian,
restricted to the $u$ term. Proceeding as before, we find

${\cal M}^{ab}=\langle
D^{*a}|H^{\pi}_{I}|D_{1}^{b}\rangle=\delta_{ab}~{}\frac{g}{\sqrt{2}}~{}\left(\frac{q^{2}}{4f_{\pi}m_{u}}\right)\sqrt{\frac{2}{3}}\,(\omega\langle
r\rangle_{2P,D\to 1S,D})$

and

$\displaystyle\Gamma(D^{0}_{1}\to\pi^{0}+D^{*0})$ $\displaystyle=$
$\displaystyle\frac{q_{1}}{6\pi}~{}g^{2}\left(\frac{q^{2}}{4f_{\pi}m_{u}}\right)^{2}\,(\omega_{1}\langle
r\rangle_{2P,D\to 1S,D})^{2}$ (59)
$\displaystyle\Gamma(D^{0}_{1}\to\pi+D^{*})$ $\displaystyle=$ $\displaystyle
3~{}\Gamma(D^{0}_{1}\to\pi^{0}+D^{*0})$ (60)

$q_{1}$ is the decay momentum, $\omega_{1}=M_{D_{1}}-M_{D^{*}}$ and we have
assumed that the $\pi D^{*}$ modes saturate the total width. The transition
radius in Eq. (59) is computed in the next Section, see Tab. 2. Using the
experimental width pdg  we find:

$g=0.63\pm 0.08$ (61)

the error is estimated from the $D_{1}^{0}$ and $D_{1}^{\pm}$ width errors and
variations in the estimated radius.

##  VI Transition radius

The transition radius for a diquarkonium was estimated in Chen:2015dig , from
the radial wave functions of a diquark-antidiquark system in a confining, QCD
potential. We solve numerically the two body, radial Schrödinger equation code
with potential and diquark mass

$\displaystyle V(r)=-\frac{\alpha_{s}}{r}+kr\qquad M_{[cq]}=1.97~{}{\rm GeV}$
(62)

Couplings are taken from lattice calculation of charmonium spectrum lattice

$\alpha_{s}=0.3\qquad k=0.15~{}{\rm GeV}^{2}$ (63)

Alternatively, Ref. Chen:2015dig  uses the parameters of the Cornell potential
cornell  or a pure confinement case:

$\displaystyle\alpha_{s}=0.47\qquad k=0.19~{}{\rm GeV}^{2}~{}({\rm Cornell})$
(64) $\displaystyle\alpha_{s}=0\qquad k=~{}0.25~{}{\rm GeV}^{2}~{}({\rm
confinement~{}only})$ (65)

For the $D_{1}\to D^{*}$ transition, we use the same potentials and
$M_{c}=1.7,~{}m_{u}=0.308$ GeV.

Transition $2P\to 1S$ | lattice, Eq. (63) | Cornell, Eq. (64) | pure confinement, Eq. (65)
---|---|---|---
$Y(4230)\to X(3872)/Z_{c}(3900)$ | $2.17$ | $1.84$ | $2.15$
$D_{1}(2420)\to D^{*}$ | $3.74$ | $3.34$ | $3.36$
Table 2: Values of the transition radius $2P\to 1S$, GeV-1, for $P$-wave
tetraquark and $D_{1}$.

Results are reported in Tab. 2

##  VII Summary

_The value of g._ The ratio of (44) to (32) depends on $g^{2}$ only:

$R_{\Gamma}=\frac{\Gamma(Y4230)\to\pi^{0}Z_{c}(3900))}{\Gamma(Y(4230)\to\gamma
X(3872))}=13.5~{}g^{2}$ (66)

Assuming that $\Gamma(Z_{c}(3900)^{0}\to(D^{*}\bar{D}+c.c)^{0})$ saturates the
$Z_{c}(3900)$ width, we obtain $g$ by comparison to the ratio of the
corresponding cross sections, Tab. 1:

$R_{\sigma}=\frac{\sigma(e^{+}e^{-}\to\pi^{0}Z_{c}(3900)^{0}\to\pi^{0}(D{\bar{D}}^{*}+c.c.)^{0})}{\sigma(e^{+}e^{-}\to\gamma
X(3872))}=14$ (67)

at $\sqrt{s}=4.226$ GeV. From $R_{\Gamma}=R_{\sigma}$, taking into account the
errors of the cross sections, we find

$g=1.0^{+0.6}_{-0.3}$ (68)

that compares well with with $g=0.6-0.7$ obtained in (57) and (61).

_The $Z_{c}(4020)$ puzzle._ The axial $\Delta L=1$ transition amplitude has a
strong dependence from the pion momentum which reflects in a steep dependence
of the rate: $\Gamma\propto q^{5}$ , see Eq. (44). The pion momentum of
$Y(4230)\to\pi Z_{c}(4020)$ implies a suppression factor $\sim 30$ with
respect to $Y(4230)\to\pi Z_{c}(3900)$, which does not seem to be supported by
the cross sections in Tab. 1.

Would it be possible that the $Z_{c}(4020)$ events come from the second peak
of the structure, $Y(4320)$? A clarification of the source of
$D^{*}\bar{D}^{*}$ events in the region and of the decay modes of $Y(4320)$
would be very useful.

_The total $Z_{c}(3900)$ width._ Following (68) and the transition radius in
Tab. 2 (lattice value), we estimate the total rate

$\Gamma(Y(4230)\to\pi Z_{c}(3900))=8^{+10}_{-4}~{}{\rm MeV}$ (69)

corresponding to a fraction $(5-36)\%$ of the total $Y(4230)$ rate, Eq. (11).

## Acknowledgements

We acknowledge a very informative exchange with Chang-Zheng Yuan on the BES
III data reported in Table 1.

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# Newform Eisenstein congruences of local origin

Dan Fretwell, Jenny Roberts

###### Abstract.

We give a general conjecture concerning the existence of Eisenstein
congruences between weight $k\geq 3$ newforms of square-free level $NM$ and
weight $k$ new Eisenstein series of square-free level $N$. Our conjecture
allows the forms to have arbitrary character $\chi$ of conductor $N$. The
special cases $M=1$ and $M=p$ prime are fully proved, with partial results
given in general. We also consider the relation with the Bloch-Kato
conjecture, and finish with computational examples demonstrating cases of our
conjecture that have resisted proof.

In memory of Lynne Walling.

##  1\. Introduction

The theory of Eisenstein congruences has a rich and beautiful history,
beginning with Ramanujan’s remarkable observation that the Fourier
coefficients $\tau(n)$ of the discriminant function:

$\Delta(z)=q\prod_{n}(1-q^{n})^{24}=\sum_{n\geq 1}\tau(n)q^{n}\in
S_{12}(\text{SL}_{2}(\mathbb{Z}))$

satisfy $\tau(n)\equiv\sigma_{11}(n)\bmod 691$ for all $n\geq 1$ (here
$\sigma_{11}(n)=\sum_{d\mid n}d^{11}$). Intuitively, this family of
congruences is explained via a congruence between two modular forms of weight
$12$; the cusp form $\Delta$ and the Eisenstein series $E_{12}$. The
significance of the modulus $691$ is that it divides the numerator of
$-\frac{B_{12}}{24}$, the constant term of $E_{12}$ (so that $E_{12}$ is a
cusp form mod $691$).

Since Ramanujan, many other congruences have been found between cusp forms and
Eisenstein series, modulo other interesting primes. For example, one can vary
the weight of the forms and find congruences whose moduli divide the
numerators of other Bernoulli numbers. The existence of such congruences has
been a key tool in the proofs of various important results in Algebraic Number
Theory, e.g. the Herbrand-Ribet Theorem, relating the $p$-divisibility of
Bernoulli numbers with the Galois module structure of
$\text{Cl}(\mathbb{Q}(\zeta_{p}))[p]$. (see [20]).

Similarly, varying the levels and characters of our forms produces even more
congruences. This time the moduli are observed to divide numerators of
generalised Bernoulli numbers and special values of local Euler factors of
Dirichlet $L$-functions. In the latter case such congruences are often
referred to as being of “local origin”. The papers [11], [12], [2], [3] and
[22] contain thorough discussions of such congruences. The existence of these
congruences is linked to special cases of the Bloch-Kato Conjecture, a far
reaching generalisation of the Herbrand-Ribet Theorem, the Analytic Class
Number Formula and the Birch Swinnerton-Dyer Conjecture. This conjecture
implies links between $p$-divisibility of special values of certain motivic
$L$-functions and $p$-torsion in certain Bloch-Kato Selmer groups.

More generally, the theory of Eisenstein congruences has been extended to
various families of automorphic form, although the landscape is still highly
conjectural [13]. Roughly speaking, if $G/\mathbb{Q}$ is a reductive group
then one expects to observe congruences between Hecke eigenvalues coming from
cuspidal automorphic representations for $G(\mathbb{A}_{\mathbb{Q}})$ and
automorphic representations that are parabolically induced from Levi subgroups
of $G$. The moduli of such congruences are also predicted to arise from
special values of certain motivic $L$-functions (related to the particular
Levi subgroup considered), in direct comparison with the Bloch-Kato
conjecture. For detailed discussion of results and conjectures in this
direction, see [1]. In the special case of $\text{GL}_{2}/\mathbb{Q}$ we
recover exactly the Eisenstein congruences mentioned above, the relevant
$L$-functions being Dirichlet $L$-functions (possibly incomplete, hence the
appearance of local Euler factors). It is expected that proving such
congruences will provide key insights into high rank cases of the Bloch-Kato
Conjecture.

In this paper we will return to the “classical” case of $\text{GL}_{2}$
Eisenstein congruences, but focus instead on the existence of newforms that
satisfy such congruences. Progress has been made on this question in the case
of trivial character ([2], [12]), and so we consider the more general case of
forms with arbitrary character of square-free conductor. To do this, we
discuss the following general conjecture and provide a full proof in the
special cases of $M=1$ and $M=p$ prime.

######  Conjecture 1.1.

Let $N,M\geq 1$ be square-free, $k>2$ and $l>k+1$ be a prime satisfying
$l\nmid NM$. Let $\psi,\phi$ be Dirichlet characters of conductors $u,v\geq
1$, satisfying $N=uv$, and set $\chi=\psi\phi$ (with a choice of lift
$\tilde{\chi}$ to modulus $NM$). There exists a newform $f\in
S_{k}^{\text{new}}(\Gamma_{0}(NM),\tilde{\chi})$ and a prime $\lambda\mid l$
of $\mathcal{O}_{f}[\phi,\psi]$ such that

$a_{q}(f)\equiv\psi(q)+\phi(q)q^{k-1}\ \text{mod }\lambda$

for all primes $q\nmid NM$ if and only if both of the following conditions
hold for some $\lambda^{\prime}|l$ in $\mathbb{Z}[\psi,\phi]$ (satisfying
$\lambda|\lambda^{\prime}$):

  1. (1)

$\text{ord}_{\lambda^{\prime}}(L(1-k,\psi^{-1}\phi)\prod_{p\in\mathcal{P}_{M}}(\psi(p)-\phi(p)p^{k}))>0$.

  2. (2)

$\text{ord}_{\lambda^{\prime}}((\psi(p)-\phi(p)p^{k})(\psi(p)-\phi(p)p^{k-2}))>0$
for each prime $p\in\mathcal{P}_{M}$.

Here, $\mathcal{P}_{M}$ is the set of primes divisors of $M$, $a_{n}(f)$ is
the $n$th Fourier coefficient of $f$ and $\mathcal{O}_{f}[\psi,\phi]$ is the
ring of integers of the smallest extension of
$K_{f}=\mathbb{Q}(\\{a_{n}(f)\\})$ containing the values of $\psi$ and $\phi$
(similarly for $\mathbb{Z}[\psi,\phi]$).

After, we consider the natural relationship between these newform congruences
and the Bloch-Kato conjecture, and give computational evidence for our
conjecture in cases where it is not known.

Acknowledgements We thank Neil Dummigan for useful discussions concerning
links between Conjecture $1.1$ and the Bloch-Kato conjecture, and for helpful
comments and suggestions for improvement.

This work forms part of the thesis of the second named author, and we are
grateful to the Heilbronn Institute for support via their PhD Studentship
program.

Finally, we wish to dedicate this work to the memory of Lynne Walling, a close
friend, mentor and collaborator of the first author, and who would have been a
PhD supervisor of the second. Lynne was a highly valued member of the
Mathematics community, whose encouragement and support towards early career
mathematicians and those from under represented groups was second to none.

##  2\. Background and notation

###  2.1. The Setup

We recap the background knowledge of modular forms that we will need, and
refer the reader to [10] for further definitions and discussions. For an
integer $N\geq 1$, define the standard congruence subgroups of
$SL_{2}(\mathbb{Z})$:

$\Gamma_{1}(N)=\left\\{\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in
SL_{2}(\mathbb{Z}):\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\equiv\begin{pmatrix}1&*\\\ 0&1\end{pmatrix}\ (\text{mod}\
N)\right\\}$ $\Gamma_{0}(N)=\left\\{\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in
SL_{2}(\mathbb{Z}):\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\equiv\begin{pmatrix}*&*\\\ 0&*\end{pmatrix}\ (\text{mod}\
N)\right\\}.$

Let $M_{k}(\Gamma_{0}(N),\chi)$ be the space of modular forms of weight $k\geq
2$, level $N$ and Dirichlet character
$\chi:(\mathbb{Z}/N\mathbb{Z})^{\times}\rightarrow\mathbb{C}^{\times}$. This
is the space of holomorphic functions $f:\mathcal{H}\rightarrow\mathbb{C}$ on
the upper half plane $\mathcal{H}$ that satisfy:

$f[\gamma]_{k}:=(cz+d)^{-k}f\left(\frac{az+b}{cz+d}\right)=\chi(d)f(z)$

for all $\gamma=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\Gamma_{0}(N)$, and
are such that $f[\alpha]_{k}$ is holomorphic at $i\infty$ for all $\alpha\in
SL_{2}(\mathbb{Z})$ (i.e. the Fourier expansion of $f[\alpha]_{k}$ is of the
form $f[\alpha]_{k}(z)=\sum_{n=0}^{\infty}a_{n}(f)q^{n}$, with $q=e^{2\pi
iz}$). Note that the weight must satisfy $\chi(-1)=(-1)^{k}$, since
$\begin{pmatrix}-1&0\\\ 0&-1\end{pmatrix}\in\Gamma_{0}(N)$.

The subspace $S_{k}(\Gamma_{0}(N),\chi)$ of cusp forms consists of the forms
such that $f[\alpha]_{k}$ has Fourier coefficient $a_{0}(f)=0$ for each
$\alpha\in SL_{2}(\mathbb{Z})$. The orthogonal complement of
$S_{k}(\Gamma_{0}(N),\chi)$ with respect to the Petersson inner product is the
Eisenstein subspace $\mathcal{E}_{k}(\Gamma_{0}(N),\chi)$. If $k>2$ (the case
of interest to us) then a natural basis of this space consists of the
normalised Eisenstein series $E_{k}^{\psi,\phi}(tz)$ for all ordered pairs of
Dirichlet characters $\phi,\psi$ of conductors $u,v$ satisfying
$\psi\phi=\chi$ and $tuv\mid N$. The Fourier expansion of $E_{k}^{\psi,\phi}$
is:

$E_{k}^{\psi,\phi}(z)=\delta(\psi)\frac{L(1-k,\psi^{-1}\phi)}{2}+\sum_{n=1}^{\infty}\sigma_{k-1}^{\psi,\phi}(n)q^{n},.$

where $\delta(\psi)=\delta_{\psi,\mathds{1}_{1}}$ (the trivial character
modulo $1$) and

$\sigma_{k-1}^{\psi,\phi}(n):=\sum_{d\mid n,d>0}\psi(n/d)\phi(d)d^{k-1}$

is a generalised power divisor sum. The Eisenstein series with $uv=N$ are
referred to as being new at level $N$.

For any given level $N$, we can also decompose $S_{k}(\Gamma_{1}(N))$ into new
and old subspaces. For any $d\mid N$ we have the map:

$\displaystyle i_{d}:S_{k}(\Gamma_{1}(N/d))^{2}$ $\displaystyle\rightarrow
S_{k}(\Gamma_{1}(N))$ $\displaystyle(f,g)$ $\displaystyle\mapsto
f+g[\alpha_{d}]_{k}$

where $\alpha_{d}=\left(\begin{smallmatrix}d&0\\\
0&1\end{smallmatrix}\right)$. Then the old subspace is:

$S_{k}^{\text{old}}(\Gamma_{1}(N))=\sum_{p\in\mathcal{P}_{N}}i_{p}(S_{k}(\Gamma_{1}(N/p))^{2}),$

where $\mathcal{P}_{N}=\\{p\text{ prime}:p\mid N\\}$. The new subspace
$S_{k}^{\text{new}}(\Gamma_{1}(N))$ is then the orthogonal complement of
$S_{k}^{\text{old}}(\Gamma_{1}(N))$ with respect to the Petersson inner
product, so that

$S_{k}(\Gamma_{1}(N))=S_{k}^{\text{old}}(\Gamma_{1}(N))\oplus
S_{k}^{\text{new}}(\Gamma_{1}(N)).$

This induces a decomposition of the space $S_{k}(\Gamma_{0}(N),\chi)$ into new
and old spaces (lifting from spaces $S_{k}(\Gamma_{0}(N/p),\chi^{\prime})$
with $p\in\mathcal{P}_{N}$ such that $p\text{ cond}(\chi)\mid N$, and taking
$\chi^{\prime}$ to be the reduction of $\chi$ mod $N/p$).

The space $M_{k}(\Gamma_{0}(N),\chi)$ comes equipped with the action of a
Hecke algebra, a commutative algebra generated by operators $T_{p}$ indexed by
primes $p$. The action of $T_{p}$ on the level of Fourier coefficients is as
follows:

$a_{n}(T_{p}(f))=a_{np}(f)+\chi(p)p^{k-1}a_{n/p}(f),$

where we take $a_{n/p}(f)=0$ if $n/p\notin\mathbb{Z}$.

The space $S_{k}(\Gamma_{0}(N),\chi)$ has a basis of eigenforms for the
operators $T_{p}$ for all $p\nmid N$. We can always normalise an eigenform $f$
so that $a_{1}(f)=1$, and in this case $T_{p}(f)=a_{p}(f)f$ for each prime
$p\nmid N$. For the subspace $S_{k}^{\text{new}}(\Gamma_{0}(N),\chi)$ we can
find a basis of newforms, eigenforms for the full Hecke algebra.

The action of the Hecke algebra on the Eisenstein subspace
$\mathcal{E}_{k}(\Gamma_{0}(N),\chi)$ is also well understood (see [10,
Proposition 5.2.3]). In particular, the normalised Eisenstein series
$E_{k}^{\psi,\phi}$ are eigenforms for all Hecke operators $T_{p}$ with
$p\nmid N$, with eigenvalues given by:

$T_{p}(E_{k}^{\psi,\phi})=\sigma_{k-1}^{\psi,\phi}(p)E_{k}^{\psi,\phi}=(\psi(p)+\phi(p)p^{k-1})E_{k}^{\psi,\phi}.$

If $E_{k}^{\psi,\phi}$ is new at level $N$ then it is an eigenform for the
full Hecke algebra and so the above holds for all $p$.

The field of definition $K_{f}=\mathbb{Q}(\\{a_{n}(f)\\})$ of an eigenform
$f\in S_{k}(\Gamma_{0}(N),\chi)$ is known to be a number field, and so has a
well defined ring of integers $\mathcal{O}_{f}$. We will often denote by
$K_{f}[\psi,\phi]$ the finite extension generated by the values of $\psi,\phi$
(i.e. roots of unity), and denote by $\mathcal{O}_{f}[\psi,\phi]$ be the
corresponding ring of integers.

By a theorem of Deligne [7], for each prime $\lambda$ of $\mathcal{O}_{f}$
there exists a continuous $\lambda$-adic Galois representation

$\rho_{f,\lambda}:G_{\mathbb{Q}}\rightarrow\text{GL}_{2}(K_{f,\lambda})$

which is unramified for $q\nmid NMl$ and satisfies

$\displaystyle\text{Tr}(\rho_{f,\lambda}(\text{Frob}_{q}))$
$\displaystyle=a_{q}(f)$
$\displaystyle\text{det}(\rho_{f,\lambda}(\text{Frob}_{q}))$
$\displaystyle=\chi(q)q^{k-1}.$

for such primes (here $\text{Frob}_{q}$ is an arithmetic Frobenius element at
$q$). By standard arguments, it is possible to conjugate $\rho_{f,\lambda}$ so
that it takes values in $\text{GL}_{2}(\mathcal{O}_{f,\lambda})$ and reduce
modulo $\lambda$ to get a continuous representation

$\overline{\rho}_{f,\lambda}:G_{\mathbb{Q}}\rightarrow\text{GL}_{2}(\mathbb{F}_{\lambda}).$

In general, the reduction depends on the choice of invariant
$\mathcal{O}_{f,\lambda}$-lattice, but the irreducible composition factors are
independent of this choice.

##  3\. A General Conjecture

From now on, $N,M\geq 1$ are fixed coprime squarefree integers and $\chi$ is a
fixed character of conductor $N$. Suppose further that $\phi,\psi$ are
characters of conductors $u,v\geq 1$ respectively satisfying $uv=N$, and set
$\psi\phi=\chi$. Then
$E_{k}^{\psi,\phi}\in\mathcal{E}_{k}(\Gamma_{0}(N),\chi)$ is new at level $N$.
Fix a lift $\tilde{\chi}$ of $\chi$ to modulus $NM$.

The following is a restatement of Conjecture $1.1$. It is a general conjecture
concerning Eisenstein congruences between newforms in
$S_{k}^{\text{new}}(\Gamma_{0}(NM),\tilde{\chi})$ and $E_{k}^{\psi,\phi}$,
providing a wide generalisation of Ramanujan’s congruence (as defined earlier,
$\mathcal{P}_{d}$ is the set of prime divisors of $d\geq 1$).

######  Conjecture 3.1.

Let $k>2$ and $l>k+1$ be a prime satisfying $l\nmid NM$. There exists a
newform $f\in S_{k}^{\text{new}}(\Gamma_{0}(NM),\tilde{\chi})$ and a prime
$\lambda\mid l$ of $\mathcal{O}_{f}[\phi,\psi]$ such that

$a_{q}(f)\equiv\psi(q)+\phi(q)q^{k-1}\ \text{mod }\lambda$

for all primes $q\nmid NM$ if and only if both of the following conditions
hold for some $\lambda^{\prime}|l$ in $\mathbb{Z}[\psi,\phi]$ (satisfying
$\lambda|\lambda^{\prime}$):

  1. (1)

$\text{ord}_{\lambda^{\prime}}(L(1-k,\psi^{-1}\phi)\prod_{p\in\mathcal{P}_{M}}(\psi(p)-\phi(p)p^{k}))>0$.

  2. (2)

$\text{ord}_{\lambda^{\prime}}((\psi(p)-\phi(p)p^{k})(\psi(p)-\phi(p)p^{k-2}))>0$
for each $p\in\mathcal{P}_{M}$.

Condition (1) of the conjecture is enough to guarantee the existence of an
eigenform in $S_{k}(\Gamma_{0}(NM),\tilde{\chi})$ satisfying the congruence
(see Theorem 3.6 below). This condition can be thought of as the analogue of
$\text{ord}_{691}\left(-\frac{B_{12}}{24}\right)>0$ in Ramanujan’s congruence,
but now allowing prime divisors of Euler factors as well as Dirichlet
$L$-values (an artifact of our Eisenstein series being lifted from level $N$
to level $NM$). Such congruences coming from divisibility of the Euler factor
(as opposed to the complete $L$-value) are often said to be of “local origin”.

Condition (2) can be thought of as measuring how “new” the modular form $f$ in
the conjecture is, i.e. if the prime $l$ only satisfied condition (2) for all
$p\in\mathcal{P}_{d}$, with $d\mid M$, then we would only expect to find a
$d$-newform $f$ satisfying the congruence condition.

We will prove the reverse implication of this Conjecture, and partial results
concerning the direct implication. The major hurdle is that the Conditions (1)
and (2) allow us to prove the existence of a $p$-newform $f$ satisfying the
congruence condition for each $p\in\mathcal{P}_{M}$, but we are currently
unable to show that these $p$-newforms can in fact be taken to the same
genuine newform.

Later, we will discuss the relationship of this conjecture with previous
results in this area, for example [11], [2], [12], as well as connections with
the Bloch-Kato Conjecture. We will also see computational examples using MAGMA
[5].

###  3.1. Initial results

To get us started, we must construct various lifts of $E_{k}^{\psi,\phi}$ to
level $MN$ having specified constant terms. The following construction and
results generalise those found in [2, Section 1.2.2]

For each $m\geq 1$ we define the operator $\alpha_{m}$, acting on complex
valued functions on the upper half plane, by $(\alpha_{m}f)(z)=f(mz)$. We then
consider the collection of Eisenstein series given by

(1)
$E_{\underline{\delta}}(z)=\left[\prod_{p\in\mathcal{P}_{M}}(T_{p}-\delta_{p})\right]\alpha_{M}E_{k}^{\psi,\phi}\in\mathcal{E}_{k}(\Gamma_{0}(NM),\tilde{\chi}),$

where $\underline{\delta}=\\{\delta_{p}\\}_{p\mid M}$ and $\delta_{p}$ is
determined by fixing an ordering:

(2) $\\{\delta_{p},\varepsilon_{p}\\}=\\{\psi(p),\phi(p)p^{k-1}\\}.$

When $M=1$ there is no choice and we define
$E_{\underline{\delta}}=E_{k}^{\psi,\phi}$.

######  Lemma 3.2.

Each Eisenstein series $E_{\underline{\delta}}$ is a normalised eigenform in
$\mathcal{E}_{k}(\Gamma_{0}(NM),\tilde{\chi})$. For each prime $p$ we have:

$T_{p}E_{\underline{\delta}}=\begin{cases}\varepsilon_{p}E_{\underline{\delta}}&\text{if
}p\in\mathcal{P}_{M}\\\
(\psi(p)+\phi(p)p^{k-1})E_{\underline{\delta}}&\text{otherwise}\\\
\end{cases}$

We can also write

(3) $E_{\underline{\delta}}=\sum_{m\mid
M}(-1)^{|\mathcal{P}_{m}|}\delta_{m}\alpha_{m}E_{k}^{\psi,\phi}.$

where $\delta_{m}=\prod_{p\in\mathcal{P}_{m}}\delta_{p}$ for each $m\mid M$.

###### Proof.

If $p\notin\mathcal{P}_{M}$ then

$\displaystyle a_{n}(T_{p}(\alpha_{M}E_{k}^{\psi,\phi}))$
$\displaystyle=a_{np}(\alpha_{M}E_{k}^{\psi,\phi})+\tilde{\chi}(p)p^{k-1}a_{n/p}(\alpha_{M}E_{k}^{\psi,\phi})$
$\displaystyle=a_{np/M}(E_{k}^{\psi,\phi})+\chi(p)p^{k-1}a_{n/pM}(E_{k}^{\psi,\phi})$
$\displaystyle=a_{n/M}(T_{p}E_{k}^{\psi,\phi})$
$\displaystyle=a_{n}(\alpha_{M}(T_{p}E_{k}^{\psi,\phi})).$

Hence, we see that
$T_{p}\alpha_{M}E_{k}^{\psi,\phi}=\alpha_{M}T_{p}E_{k}^{\psi,\phi}$. It
follows that

$\displaystyle T_{p}E_{\underline{\delta}}$
$\displaystyle=T_{p}\left[\prod_{q\in\mathcal{P}_{M}}(T_{q}-\delta_{q})\right]\alpha_{M}E_{k}^{\psi,\phi}$
$\displaystyle=\left[\prod_{q\in\mathcal{P}_{M}}(T_{q}-\delta_{q})\right]\alpha_{M}T_{p}E_{k}^{\psi,\phi}$
$\displaystyle=(\psi(p)+\phi(p)p^{k-1})E_{\underline{\delta}}$

If $p\in\mathcal{P}_{M}$ we find that:

$\displaystyle T_{p}E_{\underline{\delta}}$
$\displaystyle=T_{p}\left[\prod_{q\in\mathcal{P}_{M}}(T_{q}-\delta_{q})\right]\alpha_{M}E_{k}^{\psi,\phi}$
$\displaystyle=\left[\prod_{q\in\mathcal{P}_{M/p}}(T_{q}-\delta_{q})\right](T_{p}^{2}-\delta_{p}T_{p})\alpha_{M}E_{k}^{\psi,\phi}.$

and we must figure out the action of the final operator on
$E_{k}^{\psi,\phi}$. We do this by first proving the claim that for each
$m\mid M$:

$T_{p}\alpha_{m}E_{k}^{\psi,\phi}=\begin{cases}\alpha_{m/p}E_{k}^{\psi,\phi}&\text{if
}p\in\mathcal{P}_{m}\\\
(\psi(p)+\phi(p)p^{k-1})\alpha_{m}E_{k}^{\psi,\phi}-\chi(p)p^{k-1}\alpha_{mp}E_{k}^{\psi,\phi}&\text{if
}p\notin\mathcal{P}_{m}\end{cases}$

To prove the claim, note that

$\displaystyle a_{n}(T_{p}\alpha_{m}E_{k}^{\psi,\phi})$
$\displaystyle=a_{np}(\alpha_{m}E_{k}^{\psi,\phi})+\tilde{\chi}(p)p^{k-1}a_{n/p}(\alpha_{m}E_{k}^{\psi,\phi})$
$\displaystyle=a_{np}(\alpha_{m}E_{k}^{\psi,\phi})$
$\displaystyle=\sigma_{k-1}^{\psi,\phi}\left(\frac{np}{m}\right).$

(here, $\tilde{\chi}(p)$ vanishes since $\tilde{\chi}$ has modulus $NM$ and
$p\in\mathcal{P}_{M}$).

When $p\in\mathcal{P}_{m}$ we have
$a_{n}(T_{p}\alpha_{m}E_{k}^{\psi,\phi})=\sigma_{k-1}^{\psi,\phi}\left(\frac{np}{m}\right)=a_{n}(\alpha_{m/p}E_{k}^{\psi,\phi})$,
and when $p\notin\mathcal{P}_{m}$ we use the fact that:

(4)
$\sigma_{k-1}^{\psi,\phi}(np)+\chi(p)p^{k-1}\sigma_{k-1}^{\psi,\phi}(n/p)=(\psi(p)+\phi(p)p^{k-1})\sigma_{k-1}^{\psi,\phi}(n)$

to get

$\sigma_{k-1}^{\psi,\phi}\left(\frac{np}{m}\right)=(\psi(p)+\phi(p)p^{k-1})\sigma_{k-1}^{\psi,\phi}\left(\frac{n}{m}\right)-\chi(p)p^{k-1}\sigma_{k-1}^{\psi,\phi}\left(\frac{n}{mp}\right),$

so that

$a_{n}(T_{p}\alpha_{m}E_{k}^{\psi,\phi})=(\psi(p)+\phi(p)p^{k-1})a_{n}(\alpha_{m}E_{k}^{\psi,\phi})-\chi(p)p^{k-1}a_{n}(\alpha_{mp}E_{k}^{\psi,\phi}).$

The claim follows and so

$\displaystyle(T_{p}^{2}-\delta_{p}T_{p})\alpha_{M}E_{k}^{\psi,\phi}$
$\displaystyle=T_{p}\alpha_{M/p}E_{k}^{\psi,\phi}-\delta_{p}\alpha_{M/p}E_{k}^{\psi,\phi}$
$\displaystyle=(\psi(p)+\phi(p)p^{k-1})\alpha_{M/p}E_{k}^{\psi,\phi}-\chi(p)p^{k-1}\alpha_{M}E_{k}^{\psi,\phi}-\delta_{p}\alpha_{M/p}E_{k}^{\psi,\phi}$
$\displaystyle=\varepsilon_{p}\alpha_{M/p}E_{k}^{\psi,\phi}-\chi(p)p^{k-1}\alpha_{M}E_{k}^{\psi,\phi}$
$\displaystyle=\varepsilon_{p}(T_{p}-\delta_{p})\alpha_{M}E_{k}^{\psi,\phi}.$

Therefore, when $p\in\mathcal{P}_{M}$ we have that
$T_{p}E_{\underline{\delta}}=\varepsilon_{p}E_{\underline{\delta}}$, as
required.

Equation (3) holds by the Inclusion-Exclusion Principle and the claim. The
fact that $E_{\underline{\delta}}$ is normalised now follows, since only the
term with $m=1$ contributes to the $q$ coefficient of
$E_{\underline{\delta}}$, and $E_{k}^{\psi,\phi}$ is normalised. ∎

We now state a Proposition which allows us to determine the value of
$E_{\underline{\delta}}$ at all cusps. The proof can be found in [22,
Proposition 3.1.2] or [3, Proposition 4].

######  Proposition 3.3 (Spencer).

If $m\geq 1$ is coprime to $N$ and $\gamma=\left(\begin{smallmatrix}a&\beta\\\
b&\delta\end{smallmatrix}\right)\in SL_{2}(\mathbb{Z})$ then the constant term
of $(\alpha_{m}E_{k}^{\psi,\phi})[\gamma]_{k}$ is given by

$a_{0}((\alpha_{m}E_{k}^{\psi,\phi})[\gamma]_{k})=\begin{cases}-\frac{g(\psi\phi^{-1})}{g(\phi^{-1})}\frac{\phi^{-1}(m^{\prime}a)\psi\left(\frac{-b^{\prime}}{v}\right)}{u^{k}m^{\prime
k}}\frac{L(1-k,\psi^{-1}\phi)}{2}&\text{ if }v\mid b^{\prime}\\\
0&\text{otherwise}\end{cases}$

where $b^{\prime}=\frac{b}{gcd(b,m)}$, $m^{\prime}=\frac{m}{gcd(b,m)}$ and
$g(\phi)=\sum_{n=0}^{v-1}\phi(n)e^{\frac{2\pi in}{v}}$ is the Gauss sum of
$\phi$.

For convenience we will write:

$C_{\gamma}=-\frac{g(\psi\phi^{-1})}{g(\phi^{-1})}\frac{\phi^{-1}(a)\psi\left(\frac{-b}{v}\right)}{u^{k}}\frac{L(1-k,\psi^{-1}\phi)}{2}$

Using this and the notation in Proposition 3.3, we can write a formula for the
constant term of $E_{\underline{\delta}}[\gamma]_{k}$.

######  Corollary 3.4.

If $\gamma=\left(\begin{smallmatrix}a&\beta\\\
b&\delta\end{smallmatrix}\right)\in SL_{2}(\mathbb{Z})$ and
$M^{\prime}=\frac{M}{\text{gcd}(M,b)}$ then the constant term of
$E_{\underline{\delta}}[\gamma]_{k}$ is

$a_{0}(E_{\underline{\delta}}[\gamma]_{k})=\begin{cases}C_{\gamma}\prod_{p\in\mathcal{P}_{M^{\prime}}}(1-\delta_{p}\phi^{-1}(p)p^{-k})\prod_{p\in\mathcal{P}_{M/M^{\prime}}}(1-\delta_{p}\psi^{-1}(p))&\text{
if }v\mid b^{\prime}\\\ 0&\text{ otherwise}\end{cases}.$

###### Proof.

By Lemma 3.2, we can write $E$ as

$E_{\underline{\delta}}=\sum_{m\mid
M}(-1)^{|\mathcal{P}_{m}|}\delta_{m}\alpha_{m}E_{k}^{\psi,\phi}.$

It is clear from Proposition 3.3 that the constant term of
$E_{\underline{\delta}}$ is zero if $v\nmid b^{\prime}$ and so it suffices to
consider the case $v\mid b^{\prime}$. First we use Proposition 3.3 to evaluate
the constant term of
$(-1)^{|\mathcal{P}_{m}|}\delta_{m}\alpha_{m}E_{k}^{\psi,\phi}[\gamma]_{k}$
for a fixed $m\mid M$ as follows:

$\displaystyle(-1)^{|\mathcal{P}_{m}|+1}\frac{g(\psi\phi^{-1})}{g(\phi^{-1})}\frac{\phi^{-1}\left(m^{\prime}a\right)\psi\left(\frac{-b^{\prime}}{v}\right)}{u^{k}m^{\prime
k}}\frac{L(1-k,\psi^{-1}\phi)}{2}\delta_{m}$
$\displaystyle=C_{\gamma}(-1)^{|\mathcal{P}_{m}|}\frac{\phi^{-1}(m^{\prime})\psi^{-1}\left(\frac{m}{m^{\prime}}\right)}{m^{\prime
k}}\delta_{m}$
$\displaystyle=C_{\gamma}(-1)^{|\mathcal{P}_{m}|}\left(\prod_{p\in\mathcal{P}_{m^{\prime}}}\delta_{p}\phi^{-1}(p)p^{-k}\right)\left(\prod_{p\in\mathcal{P}_{m/m^{\prime}}}\delta_{p}\psi^{-1}(p)\right).$

It follows that the constant term of $E_{\underline{\delta}}[\gamma]_{k}$ is:

$\displaystyle a_{0}(E_{\underline{\delta}}[\gamma]_{k})$
$\displaystyle=C_{\gamma}\sum_{m\mid
M}\left[(-1)^{|\mathcal{P}_{m}|}\left(\prod_{p\in\mathcal{P}_{m^{\prime}}}\delta_{p}\phi^{-1}(p)p^{-k}\right)\left(\prod_{p\in\mathcal{P}_{m/m^{\prime}}}\delta_{p}\psi^{-1}(p)\right)\right]$
$\displaystyle=C_{\gamma}\prod_{p\in\mathcal{P}_{M^{\prime}}}(1-\delta_{p}\phi^{-1}(p)p^{-k})\prod_{p\in\mathcal{P}_{M/M^{\prime}}}(1-\delta_{p}\psi^{-1}(p))$

by the Inclusion-Exclusion Principle. ∎

###  3.2. The reverse implication

We have now constructed various Eisenstein series $E_{\underline{\delta}}$
whose constant coefficients contain terms which look somewhat like those in
Condition (1) of Conjecture 3.1. We are now in a position to use this to prove
the reverse implication of the conjecture.

######  Theorem 3.5.

The reverse implication of Conjecture 3.1 is true.

###### Proof.

Suppose that we have a newform $f\in
S_{k}^{\text{new}}(\Gamma_{0}(NM),\tilde{\chi})$ satisfying the congruence

$a_{q}(f)\equiv\psi(q)+\phi(q)q^{k-1}\ \text{mod }\lambda$

for all primes $q\nmid NM$ and for some prime $\lambda\mid l$ of
$\mathcal{O}_{f}[\psi,\phi]$ (with the assumptions $k>2$, $l>k+1$ and $l\nmid
NM$).

Attached to $f$ is the $\lambda$-adic Galois representation given by

$\rho_{f,\lambda}:G_{\mathbb{Q}}\rightarrow\text{GL}_{2}(K_{f,\lambda_{0}})\hookrightarrow\text{GL}_{2}(K_{f}[\psi,\phi]_{\lambda}),$

where the first arrow is the usual $\lambda_{0}$-adic Galois representation
attached to $f$ (with $\lambda_{0}$ the unique prime of $K_{f}$ lying below
$\lambda$). This may be conjugated to take values in
$\text{GL}_{2}(\mathcal{O}_{f}[\psi,\phi]_{\lambda})$. The congruence then
implies that $\rho_{f,\lambda}$ is residually reducible mod $\lambda$ (by
Cebotarev density and Brauer-Nesbitt, noting that $l>k>2$):

$\overline{\rho}_{f,\lambda}\sim\begin{pmatrix}\overline{\psi}&*\\\
0&\overline{\phi}\chi_{l}^{k-1}\end{pmatrix}$

so that the semisimplification is given by

(5)
$\overline{\rho}_{f,\lambda}^{ss}\sim\overline{\psi}\oplus\overline{\phi}\chi_{l}^{k-1}.$

(i.e. $\overline{\rho}_{f,\lambda}$ has composition factors
$\\{\overline{\psi},\overline{\phi}\chi_{l}^{k-1}\\}$). Here
$\chi_{l}:G_{\mathbb{Q}}\rightarrow\mathbb{F}_{l}^{\times}$ is the mod $l$
cyclotomic character.

For each $p\in\mathcal{P}_{M}$ it is well known that that the composition
factors of $\overline{\rho}_{f,\lambda}$ locally at $p$ are given by:

$\overline{\rho}^{\text{ss}}_{f,\lambda}\mid_{W_{\mathbb{Q}_{p}}}\sim\left(\mu\chi_{l}^{k/2}\oplus\mu\chi_{l}^{k/2-1}\right)\mid_{W_{\mathbb{Q}_{p}}}$

(e.g. [17, Proposition 2.8]). Here $W_{\mathbb{Q}_{p}}$ is the local Weil
group at $p$ and
$\mu:W_{\mathbb{Q}_{p}}\rightarrow\mathbb{F}_{\lambda}^{\times}$ is the
unramified character such that $\mu(\text{Frob}_{p})\equiv a_{p}(f)/p^{k/2-1}\
\text{mod }\lambda$.

This leads to the following equivalence for each $p\in\mathcal{P}_{M}$:

$\left(\mu\chi_{l}^{k/2}\oplus\mu\chi_{l}^{k/2-1}\right)\mid_{W_{\mathbb{Q}_{p}}}\sim\left(\overline{\psi}\oplus\,\overline{\phi}\chi_{l}^{k-1}\right)\mid_{W_{\mathbb{Q}_{p}}}.$

There are only two possibilities (note also that $l\neq p$ since
$p\in\mathcal{P}_{M}$):

  1. (A)

$\overline{\psi}\mid_{W_{\mathbb{Q}_{p}}}=\mu\chi_{l}^{k/2}\mid_{W_{\mathbb{Q}_{p}}}$
and
$\overline{\phi}\chi_{l}^{k-1}\mid_{W_{\mathbb{Q}_{p}}}=\mu\chi_{l}^{k/2-1}\mid_{W_{\mathbb{Q}_{p}}}$.

Evaluating at $\text{Frob}_{p}$ gives

$\displaystyle\psi(p)$ $\displaystyle\equiv\mu(p)p^{k/2}\ \text{mod }\lambda$
$\displaystyle\phi(p)p^{k-1}$ $\displaystyle\equiv\mu(p)p^{k/2-1}\ \text{mod
}\lambda,$

so that

$\psi(p)-\phi(p)p^{k}\equiv 0\ \text{mod }\lambda.$

  2. (B)

$\overline{\psi}\mid_{W_{\mathbb{Q}_{p}}}=\mu\chi_{l}^{k/2-1}\mid_{W_{\mathbb{Q}_{p}}}$
and
$\overline{\phi}\chi_{l}^{k-1}\mid_{W_{\mathbb{Q}_{p}}}=\mu\chi_{l}^{k/2}\mid_{W_{\mathbb{Q}_{p}}}$.

Evaluating at $\text{Frob}_{p}$ gives

$\displaystyle\psi(p)$ $\displaystyle\equiv\mu(p)p^{k/2-1}\ \text{mod
}\lambda$ $\displaystyle\phi(p)p^{k-1}$ $\displaystyle\equiv\mu(p)p^{k/2}\
\text{mod }\lambda,$

so that we have both of the following

$\displaystyle\psi(p)-\phi(p)p^{k-2}$ $\displaystyle\equiv 0\ \text{mod
}\lambda$ $\displaystyle\psi(p)$ $\displaystyle\equiv a_{p}(f)\bmod\lambda.$

To summarise, one of the following must hold for each $p\in\mathcal{P}_{M}$:

  1. (A)

$\psi(p)-\phi(p)p^{k}\equiv 0\ \text{mod }\lambda.$

  2. (B)

$\psi(p)-\phi(p)p^{k-2}\equiv 0\ \text{mod }\lambda$ and
$a_{p}(f)\equiv\psi(p)\ \text{mod }\lambda$.

Taking norms down to $\mathbb{Z}[\psi,\phi]$ gives Condition (2) (i.e.
divisibility by $\lambda^{\prime}$). It remains to prove that Condition (1)
holds:

$\text{ord}_{\lambda^{\prime}}\left(L(1-k,\psi^{-1}\phi)\prod_{p\in\mathcal{P}_{M}}(\psi(p)-\phi(p)p^{k})\right)>0.$

First note that this is immediate if there exists a prime
$p\in\mathcal{P}_{M}$ satisfying case (A), since $l>k+1$ and $l\nmid N$. We
assume from now on that case (B) is satisfied for each $p\in\mathcal{P}_{M}$.

Consider the Eisenstein series $E_{\underline{\delta}}$ corresponding to the
choice $\delta_{p}=\phi(p)p^{k-1}$ for each $p\in\mathcal{P}_{M}$. We claim
that for all primes $p\neq l$, the following congruence holds

$a_{p}(E_{\underline{\delta}})\equiv a_{p}(f)\ \text{mod }\lambda.$

This is true for each prime $p\in\mathcal{P}_{M}$, since by Lemma 3.2 we have

$a_{p}(E_{\underline{\delta}})=\varepsilon_{p}=\psi(p)\equiv
a_{p}(f)\bmod\lambda.$

For each prime $p\in\mathcal{P}_{N}$ the form $f$ is $p$-new, and another
comparison of local composition factors gives (e.g. [17, Proposition 2.8], see
also [3]):

$\left(\mu_{1}\chi_{l}^{(k-1)/2}\oplus\mu_{2}\chi_{l}^{(k-1)/2}\right)\mid_{W_{\mathbb{Q}_{p}}}\sim\left(\overline{\psi}\oplus\overline{\phi}\chi_{l}^{k-1}\right)\mid_{W_{\mathbb{Q}_{p}}}.$

with
$\mu_{1},\mu_{2}:W_{\mathbb{Q}_{p}}\rightarrow\mathbb{F}_{\lambda}^{\times}$
characters of conductors $1$ and $p$ satisfying

$a_{p}(f)\equiv p^{(k-1)/2}(\mu_{1}(p)+\mu_{2}(p))\ \text{mod }\lambda.$

It follows that, for such primes

$a_{p}(f)\equiv\psi(p)+\phi(p)p^{k-1}\equiv a_{p}(E_{\underline{\delta}})\
\text{mod }\lambda.$

For all other primes the claim follows from Lemma 3.2 and the assumption that
$f$ satisfies the congruence.

By the claim, and the fact that $f$ and $E_{\underline{\delta}}$ are both
normalised Hecke eigenforms, we get the following congruence for all $n$
coprime to $l$

$a_{n}(E_{\underline{\delta}})\equiv a_{n}(f)\ \text{mod }\lambda$

Applying the theta operator $\Theta=q\frac{d}{dq}$ in [21], we obtain
$\Theta(E_{\underline{\delta}})\equiv\Theta(f)\ \text{mod }\lambda$. However,
the theta operator is injective for $l>k+1$ [15, Corollary 3], and so we have
that $E_{\underline{\delta}}\equiv f\ \text{mod }\lambda$. Since $f$ is a cusp
form it must be that $E_{\underline{\delta}}$ must vanish at all cusps mod
$\lambda$. Choosing any $\gamma\in SL_{2}(\mathbb{Z})$ with lower left entry
$b$ such that $\mathcal{P}_{M}\subseteq\mathcal{P}_{b}$, we then find that

$\text{ord}_{\lambda}(a_{0}(E_{\underline{\delta}}[\gamma]_{k}))=\text{ord}_{\lambda}\left(C_{\gamma}\left(\prod_{p\in\mathcal{P}_{M}}(1-\psi^{-1}(p)\phi(p)p^{k-1})\right)\right)>0.$

by Corollary 3.4. Using the facts that $\frac{g(\psi\phi^{-1})}{g(\phi)}$,
$\phi^{-1}(a),\psi^{-1}(M)$ and $\psi\left(\frac{-b}{v}\right)$ are units in
$\mathbb{Z}[\psi,\phi]$, and that $l\nmid 2u$ by assumption, we find that:

$\text{ord}_{\lambda}\left(L(1-k,\psi^{-1}\phi)\prod_{p\in\mathcal{P}_{M}}(\psi(p)-\phi(p)p^{k-1})\right)>0.$

By assumption, we also have that $\psi(p)-\phi(p)p^{k-2}\equiv 0\bmod\lambda$
for each $p\in\mathcal{P}_{M}$, and so this implies that:

$\text{ord}_{\lambda}\left(L(1-k,\psi^{-1}\phi)\prod_{p\in\mathcal{P}_{M}}(\psi(p)-\phi(p)p^{k})\right)>0.$

Condition (1) follows again by taking the norm down to
$\mathbb{Z}[\psi,\phi]$. ∎

###  3.3. The direct implication

We now give partial results towards the direct implication of Conjecture 3.1.
First, we prove that Condition (1) guarantees the existence of an eigenform in
$S_{k}(\Gamma_{0}(NM),\tilde{\chi})$ satisfying the congruence. The following
is the necessary extension of [19, Theorem 2.10] and [22, Theorem 3.0.1].

######  Theorem 3.6.

Let $k>2$ and $\lambda^{\prime}\nmid 6NM$ be a prime of
$\mathbb{Z}[\psi,\phi]$ such that

$\text{ord}_{\lambda^{\prime}}\left(L(1-k,\psi^{-1}\phi)\prod_{p\in\mathcal{P}_{M}}(\psi(p)-\phi(p)p^{k})\right)>0.$

There exists a normalised Hecke eigenform $f\in
S_{k}(\Gamma_{1}(NM),\tilde{\chi})$ and a prime $\lambda\mid\lambda^{\prime}$
of $\mathcal{O}_{f}[\psi,\phi]$ such that for all primes $q\nmid NM$,

$a_{q}(f)\equiv\psi(q)+\phi(q)q^{k-1}\ \text{mod }\lambda.$

###### Proof.

Consider the Eisenstein series $E_{\underline{\delta}}$ corresponding to the
choice $\delta_{p}=\psi(p)$ for all $p\in\mathcal{P}_{M}$. By Corollary 3.4,
for each $\gamma\in\text{SL}_{2}(\mathbb{Z})$ the constant term of
$E_{\underline{\delta}}[\gamma]_{k}$ is either $0$ or is

$a_{0}(E_{\underline{\delta}}[\gamma]_{k})=C_{\gamma}\prod_{p\in\mathcal{P}_{M}}(1-\psi(p)\phi^{-1}(p)p^{-k})=(-1)^{|\mathcal{P}_{M}|}\frac{C_{\gamma}}{M^{k}}\prod_{p\in\mathcal{P}_{M}}(\psi(p)-\phi(p)p^{k}).$

In either case
$\text{ord}_{\lambda^{\prime}}(a_{0}(E_{\underline{\delta}}[\gamma]_{k}))>0$,
and so $E_{\underline{\delta}}$ is a cusp form mod $\lambda^{\prime}$. A
standard argument using the Deligne-Serre Lifting Lemma and Carayol’s Lemma
(e.g. [22, Theorem 3.0.1]) then gives a lift to a characteristic zero
eigenform $f\in S_{k}(\Gamma_{0}(NM),\tilde{\chi})$ satisfying $a_{q}(f)\equiv
a_{q}(E_{\underline{\delta}})\bmod\lambda$ for some prime
$\lambda|\lambda^{\prime}$ of $\mathcal{O}_{f}[\psi,\phi]$. This eigenform
satisfies the required congruence by construction. ∎

Taking $M=1$ in Theorem 3.6 gives a result of Dummigan [11, Proposition 2.1].
As remarked in the paper, the eigenform satisfying the congruence must be new
(since $\chi$ has conductor $N$). This completes the proof of Conjecture 3.1
in the case $M=1$. We will now see that the case $M=p$ prime can also be fully
proved.

######  Theorem 3.7.

If $M=p$ is prime then Conjecture 3.1 is true.

###### Proof.

Theorem 3.5 provides the reverse implication and so it suffices to prove the
direct implication. By Theorem 3.6, Condition (1) provides a level $Np$
eigenform $f_{0}\in S_{k}(\Gamma_{1}(Np),\tilde{\chi})$ and a prime
$\lambda_{0}\mid\lambda^{\prime}$ of $\mathcal{O}_{f}[\psi,\phi]$ satisfying

$a_{q}(f_{0})\equiv\psi(q)+\phi(q)q^{k-1}\ \text{mod }\lambda_{0},$

for all $q\nmid Np$. We may assume that $f_{0}$ is an oldform, otherwise we
are done. Since $\tilde{\chi}$ has conductor $N$, $f_{0}$ must be a lift of an
eigenform $f_{1}\in S_{k}^{\text{new}}(\Gamma_{0}(N),\chi)$. By the Cebotarev
density theorem, we have that
$\overline{\rho}_{f_{1},\lambda_{0}}\sim\overline{\rho}_{f_{0},\lambda_{0}}$.
As earlier, the congruence implies that
$\overline{\rho}_{f_{0},\lambda_{0}}^{\text{ss}}\sim\overline{\psi}\oplus\overline{\phi}\chi_{l}^{k-1}$.
Since $l\neq p$ we see that:

$a_{p}(f_{1})\equiv\psi(p)+\phi(p)p^{k-1}\ \text{mod }\lambda_{0}.$

By Condition (2), we also have that one of the following holds:

$\displaystyle\psi(p)$ $\displaystyle\equiv\phi(p)p^{k}\ \text{mod
}\lambda_{0}$ $\displaystyle\psi(p)$ $\displaystyle\equiv\phi(p)p^{k-2}\
\text{mod }\lambda_{0},$

so that

$a_{p}(f_{1})\equiv\psi(p)+\phi(p)p^{k-1}\equiv\begin{cases}\psi(p)(1+p^{-1})\
\text{mod }\lambda_{0}&\text{if }\psi(p)\equiv\phi(p)p^{k}\ \text{mod
}\lambda_{0}\\\ \psi(p)(1+p)\ \text{mod }\lambda_{0}&\text{if
}\psi(p)\equiv\phi(p)p^{k-2}\ \text{mod }\lambda_{0}\end{cases}$

We now claim that the following congruence condition holds:

(6) $a_{p}(f_{1})^{2}\equiv\chi(p)p^{k-2}(1+p)^{2}\ \text{mod }\lambda_{0}.$

Indeed, if $\psi(p)\equiv\phi(p)p^{k}\ \text{mod }\lambda_{0}$ then

$\displaystyle\chi(p)p^{k-2}(1+p)^{2}\equiv\psi(p)\phi(p)p^{k-2}(1+p)^{2}\equiv\psi^{2}(p)p^{-2}(1+p)^{2}$
$\displaystyle\equiv\psi^{2}(p)(1+p^{-1})^{2}$ $\displaystyle\equiv
a_{p}(f_{1})^{2}\bmod\lambda_{0}$

Alternatively, if $\psi(p)\equiv\phi(p)p^{k-2}\ \text{mod }\lambda_{0}$ then

$\displaystyle\chi(p)p^{k-2}(1+p)^{2}\equiv\psi(p)\phi(p)p^{k-2}(1+p)^{2}$
$\displaystyle\equiv\psi^{2}(p)(1+p)^{2}$ $\displaystyle\equiv
a_{p}(f_{1})^{2}\bmod\lambda_{0}.$

A well known theorem of Diamond (see [9]) now implies the existence of a
normalised $p$-newform $f\in
S_{k}^{p\text{-new}}(\Gamma_{1}(Np),\tilde{\chi})$ and a prime
$\lambda\mid\lambda_{0}$ of $\mathcal{O}_{f_{0},f}[\psi,\phi]$ satisfying

$a_{q}(f)\equiv a_{q}(f_{1})\ \text{mod }\lambda$

for all primes $q\nmid Npl$. In fact, we must have that $f\in
S_{k}^{\text{new}}(\Gamma_{0}(Np),\tilde{\chi})$, since $\tilde{\chi}$ has
conductor $N$ and $p\nmid N$. This newform satisfies the required congruence
by construction. ∎

The above argument highlights the bottleneck in trying to prove the direct
implication in general. Conditions (1) and (2) still imply the level raising
condition for each $p\mid M$, but this only allows us to find “local” newforms
satisfying the congruence, i.e. a $p$-newform for each $p|M$. There seems to
be no clear way to prove the existence of a “global” newform satisfying the
congruence.

###  3.4. Comparison with known results

In the special case of $N=1$ (so that $\chi=\mathds{1}_{1}$) Conjecture 3.1
agrees with Conjecture $4.1$ of Dummigan and Fretwell [12] and Conjecture
$3.2$ of Billerey and Menares [2].

In the special case of arbitrary square-free $N$ and $M=p$ prime, Theorem 3.6
becomes Theorem $3.0.1$ of Spencer’s thesis [22]. Newform congruences were not
explored in this thesis, and so Theorem 3.7 complements this Theorem well.

###  3.5. Low weight

For weight $k=2$, the analogue of Theorem 3.6 is expected to be true in the
case $N>1$. However, when $N=1$ the condition can fail to provide an eigenform
congruence. For example, when $N=1$ and $M=p\geq 5$ is prime then a famous
result of Mazur [18] says that eigenform congruences only arise when
$\text{ord}_{l}\left(\frac{p-1}{12}\right)>0$, as opposed to
$\text{ord}_{l}(p^{2}-1)>0$. Work of Ribet and Yoo considers results for more
general levels [23].

Conjecture 3.1 is also invalid in general for weight $k=2$. Even when an
eigenform congruence exists there may not exist a newform satisfying the
congruence, despite the fact that it is possible for Condition (2) to be
automatically satisfed. It would be interesting to see what the analogue of
Conjecture 3.1 is in this case.

It would be very interesting to see if there are analogues of our results for
weight $1$ modular forms. The existence of such eigenform congruences has been
studied for $M=p$ prime in [22], but very little seems to be known beyond
this.

##  4\. Relation with the Bloch-Kato Conjecture

In this section, we relate Conjecture 3.1 to the Bloch-Kato Conjecture.
Throughout we assume Conditions (1) and (2) and fix an eigenform $f\in
S_{k}(\Gamma_{0}(NM),\tilde{\chi})$ satisfying the congruence mod $\lambda$
(guaranteed to exist by Theorem 3.6).

As earlier, the congruence implies that the composition factors of
$\overline{\rho}_{f,\lambda}$ are given by:

$\overline{\rho}_{f,\lambda}^{\text{ss}}\sim\overline{\psi}\oplus\overline{\phi}\chi_{l}^{k-1},$

realised on the one dimensional $\mathbb{F}_{\lambda}[G_{\mathbb{Q}}]$-modules
$\mathbb{F}_{\lambda}(\psi)$ and $\mathbb{F}_{\lambda}(1-k)(\phi)$ , a Tate
twist of $\mathbb{F}_{\lambda}(\phi)$, i.e. $\text{Frob}_{p}$ acts by
multiplication by $\phi(p)p^{1-k}\bmod\lambda$ if $\lambda\nmid p$.

By a result of Ribet [20], we can choose the invariant
$\mathcal{O}_{f,\lambda}$-lattice defining $\rho_{f,\lambda}$ in such a way
that $\overline{\rho}_{f,\lambda}$ is realised on an
$\mathbb{F}_{\lambda}$-vector space $V$ such that

$0\longrightarrow\mathbb{F}_{\lambda}(1-k)(\phi)\overset{\iota}{\longrightarrow}V\overset{\pi}{\longrightarrow}\mathbb{F}_{\lambda}(\psi)\longrightarrow
0$

is a non-split extension of $\mathbb{F}_{\lambda}[G_{\mathbb{Q}}]$-modules.
Let $h:\mathbb{F}_{\lambda}(\psi)\longrightarrow V$ be an
$\mathbb{F}_{\lambda}$-linear map and fix $x\in\mathbb{F}_{\lambda}(\psi)$.
Then for each $g\in G_{\mathbb{Q}}$ we find that
$g(h(g^{-1}(x)))-h(x)\in\text{ker}(\pi)=\text{im}(\iota)$, giving a well
defined map

$\displaystyle C:G_{\mathbb{Q}}$
$\displaystyle\longrightarrow\text{Hom}(\mathbb{F}_{\lambda}(\psi),\mathbb{F}_{\lambda}(1-k)(\phi))$
$\displaystyle g$ $\displaystyle\longmapsto
C(g):x\longmapsto\iota^{-1}(g(h(g^{-1}(x)))-h(x)).$

It is easily verified that $C$ is a cocycle, whose class

$\displaystyle c=[C]$ $\displaystyle\in
H^{1}(\mathbb{Q},\text{Hom}(\mathbb{F}_{\lambda}(\psi),\mathbb{F}_{\lambda}(1-k)(\phi))$
$\displaystyle\cong
H^{1}(\mathbb{Q},\mathbb{F}_{\lambda}(1-k)(\psi^{-1}\phi)).$

is independent of the choice of $x$ and $h$, and is non-trivial since the
extension is non-split.

Consider the $\mathbb{F}_{\lambda}[G_{\mathbb{Q}}]$-module
$A_{f,\lambda}^{\psi,\phi}=(K_{f}[\psi,\phi]_{\lambda}/\mathcal{O}_{f}[\psi,\phi]_{\lambda})(1-k)(\psi^{-1}\phi)$
and let $A_{f}^{\psi,\phi}[\lambda]$ be the kernel of multiplication by
$\lambda$ (abusing notation slightly, we let $\lambda$ be a uniformiser). It
follows that

$A_{f}^{\psi,\phi}[\lambda]=\left(\frac{1}{\lambda}\mathcal{O}_{f}[\psi,\phi]_{\lambda}/\mathcal{O}_{f}[\psi,\phi]_{\lambda}\right)(1-k)(\psi^{-1}\phi)\cong\mathbb{F}_{\lambda}(1-k)(\psi^{-1}\phi)$

and so we may view $c$ as a class in
$H^{1}(\mathbb{Q},A_{f}^{\psi,\phi}[\lambda])$. The following short exact
sequence of $\mathbb{F}_{\lambda}[G_{\mathbb{Q}}]$-modules:

$0\longrightarrow
A_{f}^{\psi,\phi}[\lambda]\overset{i}{\longrightarrow}A_{f,\lambda}^{\psi,\phi}\overset{\lambda}{\longrightarrow}A_{f,\lambda}^{\psi,\phi}\longrightarrow
0$

induces a long exact sequence in Galois cohomology, a piece of which is the
following:

$H^{0}(\mathbb{Q},A_{f,\lambda}^{\psi,\phi})\overset{\delta}{\longrightarrow}H^{1}(\mathbb{Q},A_{f}^{\psi,\phi}[\lambda])\overset{i_{*}}{\longrightarrow}H^{1}(\mathbb{Q},A_{f,\lambda}^{\psi,\phi}).$

Note that since $l>k+1$ we have that $(l-1)\nmid(k-1)$, and so
$H^{0}(\mathbb{Q},A_{f,\lambda}^{\psi,\phi})$ is trivial. It follows that
$i_{*}$ is injective, so that we can lift $c$ to a non-trivial class
$c^{\prime}=i_{*}(c)\in H^{1}(\mathbb{Q},A_{f,\lambda}^{\psi,\phi})$.

The aim is to show that $c^{\prime}$ is a non-trivial element of a Bloch-Kato
Selmer group, which we now define (as in [4, §3]). First let
$B_{f,\lambda}^{\psi,\phi}=K_{f}[\psi,\phi]_{\lambda}(1-k)(\psi^{-1}\phi)$.
For a prime $q\neq l$ we define the local Bloch-Kato Selmer group attached to
$B_{f,\lambda}^{\psi,\phi}$:

$H^{1}_{f}(\mathbb{Q}_{q},B_{f,\lambda}^{\psi,\phi})=\text{ker}(H^{1}(D_{q},B_{f,\lambda}^{\psi,\phi})\longrightarrow
H^{1}(I_{q},B_{f,\lambda}^{\psi,\phi})),$

where $I_{q}\subset D_{q}\subset G_{\mathbb{Q}_{q}}$ are inertia and
decomposition subgroups at $q$. The cohomology is taken with respect to
continuous cocycles and coboundaries. Note also that the $f$ on the LHS is
standard notation and is not related to the modular form $f$. When $q=l$ the
local Bloch-Kato Selmer group of $B_{f,\lambda}^{\psi,\phi}$ is defined to be

$H^{1}_{f}(\mathbb{Q}_{l},B_{f,\lambda}^{\psi,\phi})=\text{ker}(H^{1}(D_{l},B_{f,\lambda}^{\psi,\phi})\longrightarrow
H^{1}(I_{l},B_{f,\lambda}^{\psi,\phi}\otimes_{\mathbb{Q}_{l}}B_{\text{crys}})).$

See [4, §1] for the definition of Fontaine’s ring $B_{\text{crys}}$. The
global Bloch-Kato Selmer group of $B_{f,\lambda}^{\psi,\phi}$ is then
$H^{1}_{f}(\mathbb{Q},B_{f,\lambda}^{\psi,\phi})$, the subgroup of
$H^{1}(\mathbb{Q},B_{f,\lambda}^{\psi,\phi})$ consisting of classes which have
local restriction lying in
$H^{1}_{f}(\mathbb{Q}_{q},B_{f,\lambda}^{\psi,\phi})$ for all primes $q$.

Letting $\pi:B_{f,\lambda}^{\psi,\phi}\longrightarrow
A_{f,\lambda}^{\psi,\phi}$ be the quotient map, the local Bloch-Kato Selmer
group of $A_{f,\lambda}^{\psi,\phi}$ is defined to be the pushforward
$H^{1}_{f}(\mathbb{Q}_{q},A_{f,\lambda}^{\psi,\phi})=\pi_{*}H^{1}_{f}(\mathbb{Q}_{q},B_{f,\lambda}^{\psi,\phi})$.
The global Bloch-Kato Selmer group of $A_{f,\lambda}^{\psi,\phi}$ is then
$H^{1}_{f}(\mathbb{Q},A_{f,\lambda}^{\psi,\phi})$, the subgroup of
$H^{1}(\mathbb{Q},A_{f,\lambda}^{\psi,\phi})$ consisting of classes whose
local restrictions lie in
$H^{1}_{f}(\mathbb{Q}_{q},A_{f,\lambda}^{\psi,\phi})$ for all primes $q$. Note
that, since $l\nmid 2$ we may omit $q=\infty$.

More generally, given a finite set of primes $\mathcal{P}$ with
$l\notin\mathcal{P}$, we define
$H^{1}_{\mathcal{P}}(\mathbb{Q},A_{f,\lambda}^{\psi,\phi})$ to be the subgroup
of $H^{1}(\mathbb{Q},A_{f,\lambda}^{\psi,\phi})$ consisting of classes whose
local restrictions lie in
$H^{1}_{f}(\mathbb{Q}_{q},A_{f,\lambda}^{\psi,\phi})$ for all primes
$q\notin\mathcal{P}$.

######  Proposition 4.1.

The congruence satisfied by $f$ gives the existence of a non-trivial element
$c^{\prime}\in
H^{1}_{\mathcal{P}_{NM}}(\mathbb{Q},A_{f,\lambda}^{\psi,\phi})$.

###### Proof.

We have that $\rho_{f,\lambda}$ is unramified at each $q\nmid NMl$. It follows
that restriction of $c^{\prime}$ to $H^{1}(I_{q},A_{f,\lambda}^{\psi,\phi})$
is 0 for such $q$. Then by [6, Lemma 7.4], $c^{\prime}\in
H^{1}_{f}(\mathbb{Q}_{q},A_{f,\lambda}^{\psi,\phi})$. Under the assumption
that $l>k+1$, the representation $\rho_{f,\lambda}$ is crystalline at $l$ and
we deduce that $c^{\prime}\in
H^{1}_{f}(\mathbb{Q}_{l},A_{f,\lambda}^{\psi,\phi})$, as a consequence of [8,
Proposition 2.2]. Since the necessary local conditions are satisfied, we have
that $c^{\prime}\in
H^{1}_{\mathcal{P}_{NM}}(\mathbb{Q},A_{f,\lambda}^{\psi,\phi}))$. ∎

Let $C_{k,l}^{\psi,\phi}=(\mathbb{Q}_{l}/\mathbb{Z}_{l})(1-k)(\psi^{-1}\phi)$.
Note that since $\lambda|l$, the module
$K_{f}[\psi,\phi]_{\lambda}(1-k)(\psi^{-1}\phi)$ decomposes as a direct sum of
copies of $C_{k,l}^{\psi,\phi}$. Proposition 4.1 then implies the existence of
a non-trivial element $c^{\prime}\in
H^{1}_{\mathcal{P}_{NM}}(\mathbb{Q},C_{k,l}^{\psi,\phi})$ by projection. We
now discuss how the existence of such an element agrees with the Bloch-Kato
conjecture.

Consider the partial Dirichlet $L$-value
$L_{\mathcal{P}_{NM}}(k,\psi\phi^{-1})$, i.e. with Euler factors at primes
$q\in\mathcal{P}_{NM}$ omitted. Letting $\lambda^{\prime}$ be as in Condition
(1) of Conjecture 3.1, below is a reformulation of a special case of the
$\lambda^{\prime}$-part of the Bloch-Kato conjecture (as in [8], proved in
this case by Huber and Kings in [14]).

######  Conjecture 4.2.

$\text{ord}_{\lambda^{\prime}}\left(\frac{L_{\mathcal{P}_{NM}}(k,\psi\phi^{-1})}{g(\psi\phi^{-1})(2\pi
i)^{k}}\right)=\text{ord}_{\lambda^{\prime}}\left(\frac{\text{Tam}^{0}_{\lambda}(C_{k,l}^{\psi,\phi})\\#H^{1}_{\mathcal{P}_{NM}}(\mathbb{Q},C_{k,l}^{\psi,\phi})}{\\#H^{0}(\mathbb{Q},C_{k,l}^{\psi,\phi})}\right).$

We omit the definition of the Tamagawa factor
$\text{Tam}^{0}_{\lambda}(C_{k,l}^{\psi,\phi})$, but note that it is trivial
in this case since $l>k+1$ and $\lambda^{\prime}\mid l$, by [4, Theorem
4.1.1(iii)]. We also know that $H^{0}(\mathbb{Q},C_{k,l}^{\psi,\phi}))$ is
trivial, and so

$\text{ord}_{\lambda^{\prime}}\left(\frac{L_{\mathcal{P}_{NM}}(k,\psi\phi^{-1})}{g(\psi\phi^{-1})(2\pi
i)^{k}}\right)=\text{ord}_{\lambda^{\prime}}(\\#H^{1}_{\mathcal{P}_{NM}}(\mathbb{Q},C_{k,l}^{\psi,\phi})).$

Hence if we can show that $\lambda^{\prime}$ divides the partial $L$-value
then we know there is a non-trivial element in the Bloch-Kato Selmer group.

######  Proposition 4.3.

Condition (1) of Conjecture 3.1 implies the condition

$\text{ord}_{\lambda^{\prime}}\left(\frac{L_{\mathcal{P}_{NM}}(k,\psi\phi^{-1})}{g(\psi\phi^{-1})(2\pi
i)^{k}}\right)>0.$

###### Proof.

By the functional equation for $L(s,\psi\phi^{-1})$ we have:

$\displaystyle\frac{L_{\mathcal{P}_{NM}}(k,\psi\phi^{-1})}{g(\psi\phi^{-1})(2\pi
i)^{k}}$
$\displaystyle=\frac{(-1)^{|\mathcal{P}_{NM}|}}{\phi(NM)(NM)^{k}}\frac{L(k,\psi\phi^{-1})}{g(\psi\phi^{-1})(2\pi
i)^{k}}\prod_{p\in\mathcal{P}_{NM}}(\psi(p)-\phi(p)p^{k})$
$\displaystyle=\frac{(-1)^{|\mathcal{P}_{m}|+k}}{2(k-1)!\phi(NM)(N^{2}M)^{k}}L(1-k,\psi^{-1}\phi)\prod_{p\in\mathcal{P}_{NM}}(\psi(p)-\phi(p)p^{k})$

The claim follows, since $l\nmid NM$ and $l>k+1$. ∎

The above shows that Condition (1) of Conjecture 3.1 provides a non-trivial
element $c^{\prime}\in
H^{1}_{\mathcal{P}_{NM}}(\mathbb{Q},C_{k,l}^{\psi,\phi})$, and that this is in
line with the Bloch-Kato Conjecture. Since Condition (2) is (conjecturally)
telling us that a newform $f$ of level $NM$ can be found to satisfy the
congruence, we might naively expect that the corresponding element
$c^{\prime}\in H^{1}_{\mathcal{P}_{NM}}(\mathbb{Q},C_{k,l}^{\psi,\phi})$ is
“new”, i.e. that $c^{\prime}\notin
H^{1}_{\mathcal{P}_{Nd}}(\mathbb{Q},C_{k,l}^{\psi,\phi})$ for each $d|M$ with
$d\neq M$. However, this may not be the case, since considering the Bloch-Kato
quotient for such a divisor $d$ gives:

$\displaystyle\text{ord}_{\lambda^{\prime}}\left(\frac{\\#H^{1}_{\mathcal{P}_{NM}}(\mathbb{Q},C_{k,l}^{\psi,\phi})}{\\#H^{1}_{\mathcal{P}_{Nd}}(\mathbb{Q},C_{k,l}^{\psi,\phi})}\right)$
$\displaystyle=\text{ord}_{\lambda^{\prime}}\left(\frac{L_{\mathcal{P}_{NM}}(k,\psi\phi^{-1})}{L_{\mathcal{P}_{Nd}}(k,\psi\phi^{-1})}\right)$
$\displaystyle=\text{ord}_{\lambda^{\prime}}\left(\frac{\prod_{p\in\mathcal{P}_{M/d}}(\psi(p)-\phi(p)p^{k})}{\phi(M/d)(M/d)^{k}}\right),$

revealing that new elements can only be accounted for by local divisibility
conditions of the form $\text{ord}_{\lambda^{\prime}}(\psi(p)-\phi(p)p^{k})>0$
(as in Case (A) in the proof of Theorem 3.5). It follows that the primes
$p\in\mathcal{P}_{M}$ satisfying
$\text{ord}_{\lambda^{\prime}}(\psi(p)-\phi(p)p^{k-2})>0$ and
$\text{ord}_{\lambda^{\prime}}(\psi(p)-\phi(p)p^{k})=0$ in Condition (2)
cannot contribute towards new elements.

######  Proposition 4.4.

Let $d|M$ and $d\neq M$. Then there exists a prime $p\in\mathcal{P}_{M/d}$
such that $\text{ord}_{\lambda^{\prime}}(\psi(p)-\phi(p)p^{k})>0$ if and only
if

$\text{ord}_{\lambda^{\prime}}\left(\frac{\\#H^{1}_{\mathcal{P}_{NM}}(\mathbb{Q},C_{k,l}^{\psi,\phi})}{\\#H^{1}_{\mathcal{P}_{Nd}}(\mathbb{Q},C_{k,l}^{\psi,\phi})}\right)>0.$

In particular, $\text{ord}_{\lambda^{\prime}}(\psi(p)-\phi(p)p^{k})>0$ if and
only if there exists an element $c^{\prime}_{p}\in
H^{1}_{\mathcal{P}_{NM}}(\mathbb{Q},C_{k,l}^{\psi,\phi})$ that is “$p$-new”,
i.e. $c^{\prime}_{p}\notin
H^{1}_{\mathcal{P}_{NM/p}}(\mathbb{Q},C_{k,l}^{\psi,\phi})$.

###### Proof.

This follows from the above discussion. ∎

The above should be roughly compared with the situation in Theorem 3.7 and the
proceeding discussion, where Conditions (1) and (2) alone were only able to
guarantee the existence of $p$-newforms satisfying the congruence for each
$p\in\mathcal{P}_{M}$, as opposed to a genuine newform. Based on this, we
conjecture the following.

######  Conjecture 4.5.

Let
$\mathcal{S}=\\{p\in\mathcal{P}_{M}\,|\,\text{ord}_{\lambda^{\prime}}(\psi(p)-\phi(p)p^{k})>0\\}$.
Then the class $c^{\prime}\in
H^{1}_{\mathcal{P}_{NM}}(\mathbb{Q},C_{k,l}^{\psi,\phi})$ provided by
Proposition 4.1 is “$p$-new” for each $p\in\mathcal{S}$.

Now suppose that $\text{ord}_{\lambda^{\prime}}(\psi(p)-\phi(p)p^{k})=0$ for
every $p\in\mathcal{P}_{M/d}$. In order to satisfy Condition (2) we would then
have the following conditions for each $p\in\mathcal{P}_{M/d}$:

$\displaystyle\text{ord}_{\lambda^{\prime}}(\psi(p)-\phi(p)p^{k-2})>0,$
$\displaystyle\text{ord}_{\lambda^{\prime}}(L(1-k,\psi^{-1}\phi))>0.$

Here, the situation is not fully clear (at least not to the authors). One
conclusion that can be made in this case is the following.

######  Proposition 4.6.

Let $d|M$ and $d\neq M$. If
$\text{ord}_{\lambda^{\prime}}(\psi(p)-\phi(p)p^{k-2})>0$ for each
$p\in\mathcal{P}_{M/d}$ then

$\text{ord}_{\lambda^{\prime}}\left(\frac{\\#H^{1}_{\mathcal{P}_{NM}}(\mathbb{Q},C_{k-2,l}^{\psi,\phi})}{\\#H^{1}_{\mathcal{P}_{Nd}}(\mathbb{Q},C_{k-2,l}^{\psi,\phi})}\right)\geq\\#\mathcal{P}_{M/d}.$

In particular, if $\text{ord}_{\lambda^{\prime}}(\psi(p)-\phi(p)p^{k-2})>0$
then there exists an element $c_{p}{{}^{\prime\prime}}\in
H^{1}_{\mathcal{P}_{NM}}(\mathbb{Q},C_{k-2,l}^{\psi,\phi})$ that is “$p$-new”,
i.e. $c_{p}{{}^{\prime\prime}}\notin
H^{1}_{\mathcal{P}_{NM/p}}(\mathbb{Q},C_{k-2,l}^{\psi,\phi})$.

###### Proof.

This follows by considering the Bloch-Kato quotient of weight $k-2$:

$\displaystyle\text{ord}_{\lambda^{\prime}}\left(\frac{\\#H^{1}_{\mathcal{P}_{NM}}(\mathbb{Q},C_{k-2,l}^{\psi,\phi})}{\\#H^{1}_{\mathcal{P}_{Nd}}(\mathbb{Q},C_{k-2,l}^{\psi,\phi})}\right)$
$\displaystyle=\text{ord}_{\lambda^{\prime}}\left(\frac{L_{\mathcal{P}_{NM}}(k-2,\psi\phi^{-1})}{L_{\mathcal{P}_{Nd}}(k-2,\psi\phi^{-1})}\right)$
$\displaystyle=\text{ord}_{\lambda^{\prime}}\left(\frac{\prod_{p\in\mathcal{P}_{M/d}}(\psi(p)-\phi(p)p^{k-2})}{\phi(M/d)(M/d)^{k-2}}\right).$

∎

Alternatively, the divisibility condition in the above Proposition implies
Conditions (1) and (2) for weight $k-2$. By Theorem 3.6 there exists a
congruence between the eigenvalues of an eigenform $g\in
S_{k-2}(\Gamma_{0}(NM/d),\tilde{\chi})$ and the eigenvalues of
$E_{k-2}^{\psi,\phi}$ (at “good” primes). By a similar argument to earlier,
this supplies a non-trivial element $c^{{}^{\prime\prime}}\in
H^{1}_{\mathcal{P}_{NM/d}}(\mathbb{Q},C_{k-2,l}^{\psi,\phi})$. Conjecture 3.1
would let us take $g$ to be a newform, and so we are led to conjecture the
following.

######  Conjecture 4.7.

Let
$\mathcal{S}=\\{p\in\mathcal{P}_{M}\,|\,\text{ord}_{\lambda^{\prime}}(\psi(p)-\phi(p)p^{k-2})>0\\}$.
Then the class $c^{{}^{\prime\prime}}\in
H^{1}_{\mathcal{P}_{NM/d}}(\mathbb{Q},C_{k-2,l}^{\psi,\phi})$ constructed
above is $p$-new for each $p\in\mathcal{S}$.

In summary, we have considered the two different ways in which Conditions (1)
and (2) can hold. By Bloch-Kato, each implies the existence of a collection
“$p$-new” elements in a certain Bloch-Kato Selmer group (i.e. Propositions 4.4
and 4.6). Further, we expect that both collections of $p$-new elements should
be explained by the existence of two “new” elements, each arising from a
newform congruence implied by Conjecture 3.1 (i.e. Conjectures 4.5 and 4.7).
In the case of $M=p$ prime, we remark that Theorem 3.7 implies the truth of
these conjectures, but for more general square-free $M$, little seems to be
known.

##  5\. Examples

Below, we give brief computational examples of Conjecture 3.1, using data
provided by the LMFDB databse [16]. First we consider examples demonstrating
Theorem 3.7.

######  Example 5.1.

Take $N=5$, $M=2$ and $k=8$. Also let $\psi=\mathds{1}$ and
$\phi=\left(\frac{\cdot}{5}\right)$ (so that $\chi=\phi$). The only prime
$\lambda^{\prime}$ of $\mathbb{Z}[\psi,\phi]=\mathbb{Z}$ satisfying the
conditions of Theorem 3.7 is $\lambda^{\prime}=257$.

Indeed, the newform $f\in S_{8}^{\text{new}}(\Gamma_{0}(10),\tilde{\chi})$
with LMFDB label [10.8.b.a] satisfies the congruence

$a_{q}(f)\equiv 1+\left(\frac{q}{5}\right)q^{7}\bmod\lambda$

for all $q\neq 2,5$ and for some fixed prime $\lambda\mid\lambda^{\prime}$ of
$\mathcal{O}_{f}[\psi,\phi]=\mathcal{O}_{f}$ (the ring of integers of the
quartic field generated by a root of $x^{4}-15x^{2}+64$).

######  Example 5.2.

Take $N=7$, $M=2$ and $k=7$. Also let $\psi=\mathds{1}$ and $\phi$ be the
primitive mod $7$ character satisfying $\phi(3)=\zeta_{6}$ (so that
$\chi=\phi$). The only prime $\lambda^{\prime}$ of
$\mathbb{Z}[\psi,\phi]=\mathbb{Z}[\zeta_{6}]$ satisfying the conditions of
Theorem 3.6 is $\lambda^{\prime}=\langle 337,\zeta_{6}+128\rangle$ (lying
above $l=337$).

Indeed, the newform $f\in S_{7}^{\text{new}}(\Gamma_{0}(14),\tilde{\chi})$
with LMFDB label [14.7.d.a] satisfies the congruence

$a_{q}(f)\equiv 1+\phi(q)q^{6}\bmod\lambda$

for all $q\neq 2,7$ and for some fixed prime $\lambda\mid\lambda^{\prime}$ of
$\mathcal{O}_{f}[\psi,\phi]=\mathcal{O}_{f}[\zeta_{6}]$ (here
$\mathcal{O}_{f}$ is the ring of integers of a degree $8$ number field).

We finish with an example demonstrating Conjecture 3.1 in a case where $M$ is
composite.

######  Example 5.3.

Take $N=7,M=6$ and $k=6$. Let $\psi=\mathds{1}$ and $\phi$ be the primitive
mod $7$ character satisfying $\phi(3)=\zeta_{3}^{2}$ (so that $\chi=\phi$).
The only prime $\lambda^{\prime}$ of
$\mathbb{Z}[\psi,\phi]=\mathbb{Z}[\zeta_{6}]$ satisfying the conditions of
Theorem 3.6 is $\lambda^{\prime}=\langle 73,\zeta_{6}+64\rangle$ (lying above
$l=73$).

Indeed, the newform $f\in S_{6}^{\text{new}}(\Gamma_{0}(42),\tilde{\chi})$
with LMFDB label [42.6.e.c] satisfies the congruence:

$a_{q}(f)\equiv 1+\phi(q)q^{6}\bmod\lambda$

for all primes $q\neq 2,7$ and for a fixed prime $\lambda\mid\lambda^{\prime}$
of $\mathcal{O}_{f}[\psi,\phi]=\mathcal{O}_{f}[\zeta_{6}]$ (here
$\mathcal{O}_{f}$ is the ring of integers of a degree $4$ number field).

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