$$\lambda_{\pi,j}=:\frac{1}{4}-\nu_{\pi,j}^2\in{\mathbb{R}}.$$ Here $\nu_{\pi,j}\in\frac{1}{2}{\mathbb{Z}}$ if $\pi_{\infty,j}$ belongs to the discrete series, and by [@KSh] we have $$\label{boundLaplace}
\nu_{\pi,j}\in i{\mathbb{R}}\cup[-\theta,\theta],\qquad \theta:=1/9,$$ if $\pi_{\infty,j}$ belongs to the principal series or the complementary series. We shall choose $\nu_{\pi,j}$ so that $\Re
\nu_{\pi,j}{\geqslant}0$ and $\Im \nu_{\pi,j} {\geqslant}0$. For notational simplicity we write $$\label{short}
\begin{aligned}
\lambda_\pi&:=(\lambda_{\pi,j})_{j=1}^d\in{\mathbb{R}}^d,\\
\nu_\pi&:=(\nu_{\pi,j})_{j=1}^d\in{\mathbb{R}}^d,\end{aligned}\qquad
\begin{aligned}
\tilde\lambda_\pi&:=(1+|\lambda_{\pi,j}|)_{j=1}^d\in{\mathbb{R}}_{>0}^d;\\
\tilde\nu_\pi&:=(1+|\nu_{\pi,j}|)_{j=1}^d\in{\mathbb{R}}_{>0}^d;
\end{aligned}$$ in particular, $${\mathcal{N}}\tilde{\lambda}_{\pi} = \prod_{j=1}^d
(1+|\lambda_{\pi,j}|)\qquad\text{and}\qquad{\mathcal{N}}\tilde{\nu}_{\pi} =
\prod_{j=1}^d (1+|\nu_{\pi,j}|).$$

Let ${\mathfrak{c}}_\omega\subseteq{\mathfrak{o}}$ denote the conductor of the central character $\omega$, and for a nonzero ideal ${\mathfrak{c}}\subseteq
{\mathfrak{c}}_\omega$ let $$V_{\pi}({\mathfrak{c}}) :=\left\{\phi \in V_{\pi} \mid \text{$\phi(g\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)) =
 \omega_{{\mathfrak{c}}}(d)\phi(g)$ for all
 $g\in\operatorname{GL}_2(\mathbb{A})$ and $\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right) \in
 {\mathcal{K}}({\mathfrak{c}})$}\right\},$$ where $$\omega_{{\mathfrak{c}}}(x) :=
\prod_{{\mathfrak{p}}\mid{\mathfrak{c}}} \omega_{\mathfrak{p}}(x),\qquad x\in\mathbb{A}^\times.$$ For ${\mathfrak{c}}\subseteq{\mathfrak{c}}'\subseteq{\mathfrak{c}}_\omega$ we have $V_\pi({\mathfrak{c}}')\subseteq V_\pi({\mathfrak{c}})$, because[^6] $\omega_{\mathfrak{c}}(d)=\omega_{{\mathfrak{c}}'}(d)$ for $\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right) \in
 {\mathcal{K}}({\mathfrak{c}})$. We define the conductor ${\mathfrak{c}}_{\pi}$ of $\pi$ as the largest ideal ${\mathfrak{c}}\subseteq{\mathfrak{c}}_\omega$ such that $V_{\pi}({\mathfrak{c}}) \neq \{0\}$ (cf.[@Ca Theorem 1] and [@Mi Corollary 2]). Analogously, for a prime ${\mathfrak{p}}$ and a nonzero ideal ${\mathfrak{c}}\subseteq
{\mathfrak{c}}_{\omega_{{\mathfrak{p}}}}$ we define $V_{\pi_{{\mathfrak{p}}}}({\mathfrak{c}})$ and the local conductor ${\mathfrak{c}}_{\pi_{{\mathfrak{p}}}}$. Note that ${\mathfrak{c}}_{\pi_{{\mathfrak{p}}}} =
{\mathfrak{c}}_{\pi} {\mathfrak{o}}_{{\mathfrak{p}}}$. Finally, we define the analytic conductor of $\pi$ (cf. [@IS]) as $$\label{analyticconductor}
C(\pi):=({\mathcal{N}}{\mathfrak{c}}_\pi)({\mathcal{N}}\tilde\lambda_\pi).$$

For any nonzero ideals ${\mathfrak{t}},{\mathfrak{c}}\subseteq{\mathfrak{o}}$ such that ${\mathfrak{t}}{\mathfrak{c}}_\pi\mid{\mathfrak{c}}$ there is an isometric embedding of complex vector spaces $$\label{defshift}
R_{\mathfrak{t}}:V_\pi({\mathfrak{c}}_\pi)\hookrightarrow V_\pi({\mathfrak{c}}),\qquad\qquad
(R_{\mathfrak{t}}\phi)(g):= \phi\left(g\left(\begin{smallmatrix}t^{-1}&0\\0&1\end{smallmatrix}\right)\right),$$ where 