
\begin{lemme}
On suppose $n_\chi>0$. Alors:
\begin{align*}
& \rightarrow \text { Si } n_\psi<n_{\chi}, \quad G(\chi, \psi)=0 \\
& \rightarrow \text{Si } n_\psi=n_{\chi}, \quad|G(\chi, \psi)|_{\infty}^2=\frac{|\OO / \mathcal{D} |^{-1 / 2} }{1-q^{-1}} \operatorname{Vol}\left(U_{n_\psi}, dx^{\times}\right) \\
& \rightarrow \text{Si }n_\psi>n_{\chi}, \quad|G(\chi, \psi)|_{\infty}^2=\frac{|\OO / \mathcal{D} |^{-1 / 2} }{1-q^{-1}}\left(\operatorname{Vol}\left(U_{n_\psi}, dx^{\times}\right)-q^{-1} \operatorname{Vol}\left(U_{n_{\psi^{-1}}}, dx^{\times}\right)\right)
\end{align*}
\end{lemme}


\begin{lemme}
On suppose $n_\chi>0$. Alors:
\begin{align*}
& \rightarrow \text { Si } n_\psi<n_{\chi}, \quad G(\chi, \psi)=0 \\
& \rightarrow \text{Si } n_\psi=n_{\chi}, \quad|G(\chi, \psi)|_{\infty}^2=\frac{|\OO / \mathcal{D} |^{-1 / 2} }{1-q^{-1}} \operatorname{Vol}\left(U_{n_\psi}, dx^{\times}\right) \\
& \rightarrow \text{Si }n_\psi>n_{\chi}, \quad|G(\chi, \psi)|_{\infty}^2=\frac{|\OO / \mathcal{D} |^{-1 / 2} }{1-q^{-1}}\left(\operatorname{Vol}\left(U_{n_\psi}, dx^{\times}\right)-q^{-1} \operatorname{Vol}\left(U_{n_{\psi^{-1}}}, dx^{\times}\right)\right)
\end{align*}
\end{lemme}
