Poincaré series $F_m(\tau,s)$.

\[fourier1\] The automorphic Green function $\tilde\Phi_{mp^2}(z_1,z_2,s)$ associated to $\tilde T_m$ has the Fourier expansion $$\begin{aligned}
&\tilde\Phi_{mp^2}(z_1,z_2,s)\\
&=\left(\frac{p}{\pi}\right)^{s-1/2}\Gamma(s-1/2)L(2s-1,\epsilon_p)
b_m(0,s)(y_1 y_2)^{1-s}\\
&\phantom{=}{}+\sqrt{p} \sum_{\substack{\nu\in \partial_F^{-1}\\\nu\nu'\neq 0}}\sum_{n\geq 1}
 \frac{\epsilon_p(n)}{n} b_m(p\nu\nu',s){\mathcal{K}}_{s}(2\pi n\nu y_1) {\mathcal{K}}_{s}(2\pi n\nu' y_2)e(\nu n x_1+\nu'n x_2)\\
&\phantom{=}{}+\sqrt{p} \sum_{\substack{ \lambda\in\partial_F^{-1}\\ {\operatorname{N}}(\lambda)=-m/p}} \sum_{n\geq 1}
\frac{\epsilon_p(n)}{n}\big(
2{\mathcal{I}}_{s}(2\pi n \beta(\lambda y_1,\lambda' y_2))+{\mathcal{K}}_{s}(2\pi n \beta(\lambda y_1,\lambda' y_2))\big)\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad \times {\mathcal{K}}_{s}(2\pi n \alpha(\lambda y_1,\lambda' y_2)) e(n\lambda
x_1+n\lambda' x_2).\end{aligned}$$ If converges normally for $y_1 y_2>mp$. Here $b_m(n,s)$ denote the Fourier coefficients of the Poincaré series $F_m(\tau,s)$, and $L(s,\epsilon_p)$ the Dirichlet $L$-function corresponding to $\epsilon_p$.

We use , Lemma \[lphi0\], and Lemma \[lHA\] to compute the Fourier expansion. The constant term is given by $$\begin{aligned}
& 2  \sum_{a=1}^{\infty} \tilde G_a(mp^2,0)b_s^{mp/a^2}(0,y_1,y_2)\\
&=\frac{\pi \Gamma(s-1/2)^2}{ \sqrt{p}\,\Gamma(2s)}(4mp)^s (y_1 y_2)^{1-s} \sum_{a=1}^{\infty} \tilde G_a(mp^2,0)
a^{-2s}.\end{aligned}$$ In view of Lemma \[GaHb\] this is equal to $$\begin{aligned}
\nonumber &\frac{\pi \Gamma(s-1/2)^2}{ \sqrt{p}\,\Gamma(2s)}(4mp)^s (y_1 y_2)^{1-s} \sum_{a=1}^{\infty} \sum_{r|a}
\epsilon_p(r) H_{a/r}\left( m,0\right)
 a^{1-2s}\\
\nonumber &=\frac{\pi \Gamma(s-1/2)^2}{ \sqrt{p}\,\Gamma(2s)}(4mp)^s (y_1 y_2)^{1-s} \sum_{c=1}^{\infty}
\sum_{r=1}^\infty \epsilon_p(r) H_{c}\left( m,0 \right)
 (cr)^{1-2s}\\
\label{h1} &=\frac{\pi \Gamma(s-1/2)^2}{ \sqrt{p}\,\Gamma(2s)}(4mp)^s (y_1 y_2)^{1-s} L(2s-1,\epsilon_p)
\sum_{c=1}^{\infty} H_{c}\left( m,0\right) c^{1-2s}.\end{aligned}$$ By means of the formula for the constant coefficient $b_m(0,s)$ of $F_m(\tau,s)$ and the duplication formula $\Gamma(s)\Gamma(s+\frac{1}{2})=2^{1-2s}\sqrt{\pi}\Gamma(2s)$ we see that the constant term equals: $$\begin{aligned}
\left(\frac{p}{\pi}\right)^{s-1/2}\Gamma(s-1/2)L(2s-1,\epsilon_p) b_m(0,s)(y_1 y_2)^{1-s}.\end{aligned}$$ We now consider the 