${\varepsilon}_\pi:\{\pm 1\}^d\to\{\pm 1\}$ depending only on $\pi$ such that every $\phi \in V_{\pi, q}({\mathfrak{c}})$ has a Fourier–Whittaker expansion (cf. [@KM (2.11), (3.8)]) $$\label{fourierwhittaker}
\phi\left(\begin{pmatrix} y & x\\ 0 & 1\end{pmatrix}\right) =
\rho_{\phi,0}(y) + \sum_{r \in K^\times} \rho_{\phi}(r
y_{\text{fin}}){\varepsilon}_\pi(\operatorname{sgn}(r y_\infty))\tilde{W}_{q/2,
\nu_{\pi}}(r y_{\infty}) \psi(rx)$$ for $y = y_{\infty}\times y_{\text{fin}}\in \mathbb{A}^{\times}$, $x
\in \mathbb{A}$. Note that for any $y_\text{fin}\in\mathbb{A}_\text{fin}^\times$ the coefficient $\rho_\phi (y_\text{fin})$ only depends on the fractional ideal represented by $y_\text{fin}$ and it is nonzero only if this ideal is integral. The normalization of $\tilde{W}_{q/2, \nu_{\pi}}$ is further justified by the fact that these coefficients remain unchanged if $\phi$ is replaced by any of its nonzero Maa[ß]{} shifts.

If $(\pi, V_{\pi})$ is one of the right $\operatorname{GL}_2(\mathbb{A})$-spaces $V_\chi$ with $\chi\in X$ in , then the expansion only contains the constant term $\rho_0(y)$.

Let us now assume that $(\pi, V_{\pi})$ is one of the right $\operatorname{GL}_2(\mathbb{A})$-spaces $V_{\pi}$ with $\pi\in
\mathcal{C}_{\omega}$ in , so that $\rho_0(y)=0$. The finer structure of the coefficients $\rho_\phi$ can be revealed by the theory of Hecke operators, as developed by Miyake [@Mi] (see also [@Sh1 Section 2] and [@KM Section 2]). By we can decompose any vector $\phi \in V_{\pi,
q}({\mathfrak{c}})$ as $$\phi=\sum_{{\mathfrak{t}}\mid{\mathfrak{c}}{\mathfrak{c}}_{\pi}^{-1}} R_{\mathfrak{t}}\phi_{\mathfrak{t}},$$ where each $\phi_{\mathfrak{t}}$ lies in 