$\nu$-th Fourier coefficient of $\tilde\Phi_{mp^2}(z_1,z_2,s)$ for $\nu\in \partial_F$ with $\nu\nu'>0$. It is given by $$\begin{aligned}
& 2  \sum_{a=1}^{\infty} \tilde G_a(mp^2,\nu)b_s^{mp/a^2}(\nu,y_1,y_2)\\
&=2\pi  \sum_{a=1}^{\infty} \frac{1}{a}\tilde
G_a(mp^2,\nu)\sqrt{\frac{m}{\nu\nu'}}\,I_{2s-1}\left(\frac{4\pi}{a}\sqrt{mp|\nu\nu'|}\right){\mathcal{K}}_{s}(2\pi \nu
y_1) {\mathcal{K}}_{s}(2\pi \nu' y_2).\end{aligned}$$ In view of Lemma \[GaHb\] this is equal to: $$\begin{aligned}
&2\pi  \sum_{a=1}^{\infty}  \sum_{\substack{r|\nu \\ r|a}}
\epsilon_p(r) H_{a/r}\left(m, \frac{p\nu\nu'}{ r^2}\right)\sqrt{\frac{m}{\nu\nu'}}\,I_{2s-1}\left(\frac{4\pi}{a}\sqrt{mp|\nu\nu'|}\right){\mathcal{K}}_{s}(2\pi \nu y_1) {\mathcal{K}}_{s}(2\pi \nu' y_2)\\
&=2\pi  \sqrt{p}  \sum_{\substack{r|\nu }} \frac{\epsilon_p(r)}{r}
\sqrt{\frac{mr^2}{p\nu\nu'}}\sum_{c=1}^{\infty}   H_{c}\left(m, \frac{p\nu\nu'}{ r^2}\right)
I_{2s-1}\left(\frac{4\pi}{cr}\sqrt{mp|\nu\nu'|}\right){\mathcal{K}}_{s}(2\pi \nu y_1) {\mathcal{K}}_{s}(2\pi \nu' y_2).\end{aligned}$$ By Theorem \[fourierF\] we finally find for the $\nu$-th coefficient: $$\begin{aligned}
&  \sqrt{p}  \sum_{\substack{r|\nu }} \frac{\epsilon_p(r)}{r} b_m(p\nu\nu'/r^2,s){\mathcal{K}}_{s}(2\pi \nu y_1)
{\mathcal{K}}_{s}(2\pi \nu' y_2).\end{aligned}$$

In the same way one can show that for $\nu\in \partial_F^{-1}$ with $\nu\nu'<0$ the $\nu$-th coefficient of $\tilde\Phi_{mp^2}(z_1,z_2,s)-\tilde\Phi_{mp^2}^0(z_1,z_2,s)$ is equal to $$\begin{aligned}
&  \sqrt{p}  \sum_{\substack{r|\nu }} \frac{\epsilon_p(r)}{r} b_m(p\nu\nu'/r^2,s){\mathcal{K}}_{s}(2\pi \nu y_1) {\mathcal{K}}_{s}(2\pi \nu' y_2)\\
&+ \sqrt{p}  \sum_{\substack{r|\nu }} \frac{\epsilon_p(r)}{r} \delta_{-m,p\nu\nu'/r^2}{\mathcal{K}}_{s}(2\pi \nu y_1)
{\mathcal{K}}_{s}(2\pi \nu' y_2).\end{aligned}$$ Here the extra contribution with the Kronecker delta comes from the Kronecker delta $\delta_{-m,n}$ in the formula for $b_m(n,s)$ with $n<0$.

If we put the above contributions together, and use in addition the formula for $\tilde\Phi_{mp^2}^0$ of Lemma \[lphi0\], we find $$\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_{\substack{r|\nu }} \frac{\epsilon_p(r)}{r} b_m(p\nu\nu'/r^2,s){\mathcal{K}}_{s}(2\pi \nu y_1) {\mathcal{K}}_{s}(2\pi \nu' y_2)e(\nu x_1+\nu' x_2)\\
&\phantom{=}{}+\sqrt{p}  \sum_{\substack{\nu\in \partial_F^{-1}\\\nu\nu'< 0}} 