$V_{\pi, q}({\mathfrak{c}}_\pi)$. By we infer $$\rho_{\phi}(\mathfrak{m})=\sum_{{\mathfrak{t}}\mid{\mathfrak{c}}{\mathfrak{c}}_{\pi}^{-1}}
\rho_{R_{\mathfrak{t}}\phi_{\mathfrak{t}}}(\mathfrak{m}) =\sum_{{\mathfrak{t}}\mid{\mathfrak{c}}{\mathfrak{c}}_{\pi}^{-1}}
\rho_{\phi_{\mathfrak{t}}}(\mathfrak{m}{\mathfrak{t}}^{-1}),\qquad\mathfrak{m}\subseteq{\mathfrak{o}},$$ so that we can focus our attention to the case when ${\mathfrak{c}}={\mathfrak{c}}_\pi$, i.e. when $\phi\in V_{\pi, q}({\mathfrak{c}}_\pi)$ is a newform. For each nonzero ideal $\mathfrak{m} \subseteq {\mathfrak{o}}$ the Hecke operator $T_{{\mathfrak{c}}_\pi}(\mathfrak{m})$ acts on $V_{\pi}({\mathfrak{c}}_{\pi})$ by a scalar $\lambda_{\pi}(\mathfrak{m})$. The function $\lambda_{\pi}$ satisfies $$\label{hecke}
 \lambda_{\pi}(\mathfrak{m})\lambda_{\pi}(\mathfrak{n}) =
 \sum_{{\mathfrak{a}}\mid\gcd(\mathfrak{m},\mathfrak{n})}
 \omega_{\pi}({\mathfrak{a}})\lambda_{\pi}(\mathfrak{m}\mathfrak{n}{\mathfrak{a}}^{-2}),$$ and $$\label{hecke+}
 \lambda_{\pi}(\mathfrak{m}) = \omega_{\pi}(\mathfrak{m})
 \bar{\lambda}_{\pi}(\mathfrak{m}),\qquad\gcd(\mathfrak{m},{\mathfrak{c}}_\pi)={\mathfrak{o}},$$ where $\omega_\pi:I(K)\to{\mathbb{C}}$ is defined as follows: if ${\mathfrak{a}}\in
I(K)$ is coprime to ${\mathfrak{c}}_{\pi}$ then $\omega_{\pi}({\mathfrak{a}}): =
\omega(a)$ where $a \in \mathbb{A}_{\text{fin}}^{\times}$ is any finite idele representing ${\mathfrak{a}}$ with $a_{\mathfrak{p}}= 1$ for ${\mathfrak{p}}\mid
{\mathfrak{c}}_{\pi}$, otherwise $\omega_{\pi}({\mathfrak{a}}): = 0$. In particular, $\lambda_{\pi}$ is multiplicative on the set of nonzero integral ideals. The non-archimedean analogue of is $$\label{boundlambda}
 \lambda_{\pi}(\mathfrak{m}) \ll_{{\varepsilon}} (N\mathfrak{m})^{\theta +{\varepsilon}},\qquad
 \theta:=1/9,$$ for any ${\varepsilon}> 0$, see [@KSh]. It follows, as stated after , that each $V_{\pi, q}({\mathfrak{c}}_{\pi})$ is at most one-dimensional, and in fact $$\rho_{\phi}(\mathfrak{m}) = \frac{\lambda_{\pi}(\mathfrak{m})}{\sqrt{{\mathcal{N}}\mathfrak{m}}}\rho_\phi({\mathfrak{o}}),
 \qquad\mathfrak{m}\subseteq{\mathfrak{o}},\qquad \phi\in
V_{\pi, q}({\mathfrak{c}}_{\pi}).$$ To maintain this identity we define $\lambda_\pi({\mathfrak{a}})$ to be zero for any nonintegral ${\mathfrak{a}}\in I(K)$. Comparing with we see that for $\phi\in V_{\pi,
q}({\mathfrak{c}}_{\pi})$ we have $$\label{fouriersimple}
\phi\left(\left(\begin{matrix} y & x\\ 0 &
1\end{matrix}\right)\right) = \sum_{r \in K^\times}
\frac{\lambda_{\pi}(r y_{\text{fin}})}{\sqrt{{\mathcal{N}}(r y_{\text{fin}})}}
W_{\phi}(r y_{\infty}) \psi(rx),\qquad y\in \mathbb{A}^{\times},\ \
x 