y_1,\lambda' y_2))\\
{}\times {\mathcal{K}}_{s}(2\pi n \alpha(\lambda y_1,\lambda' y_2))e(n\lambda x_1+n\lambda' x_2).\end{gathered}$$

By definition we have $$\begin{aligned}
\tilde\Phi_{mp^2}^0(z_1,z_2,s) &= \sum_{\substack{b\in {\mathbb{Z}},\,\lambda\in \partial_F\\ \lambda\lambda'=-mp}}
\epsilon_p(b)
Q_{s-1}\left(1+\frac{ |\lambda z_1+\lambda' z_2+b|^2}{ 2y_1 y_2 mp }\right)\\
&= \sum_{\substack{b\,(p),\,\lambda\in \partial_F^{-1}\\ \lambda\lambda'=-m/p}} \epsilon_p(b) \sum_{b'\in {\mathbb{Z}}}
Q_{s-1}\left(1+\frac{ |\lambda z_1+\lambda' z_2+b'+b/p|^2}{ 2y_1 y_2 m/p }\right).\end{aligned}$$ We find that $$\begin{aligned}
\label{Phi0h}
\tilde\Phi_{mp^2}^0(z_1,z_2,s) = \sum_{\substack{b\,(p)}} \sum_{\substack{\lambda\in \partial_F^{-1}\\
\lambda\lambda'=-m/p}}\epsilon_p(b) h_{\alpha(\lambda y_1, \lambda' y_2), \beta(\lambda y_1, \lambda'
y_2)}(\lambda x_1+ \lambda' x_2+b/p),\end{aligned}$$ where $$h_{\alpha,\beta}(x) = \sum_{b\in{\mathbb{Z}}}  Q_{s-1}\left(\frac{(x+b)^2+\alpha^2+\beta^2}{2\alpha\beta}\right).$$ By [@Br1] Lemma 1, for $\alpha>\beta>0$, the function $h_{\alpha,\beta}(x)$ has the Fourier expansion $$h_{\alpha,\beta}(x)= \frac{2\pi}{2s-1} \alpha^{1-s}\beta^{s} +\sum_{n\in{\mathbb{Z}}-\{0\}}\frac{1}{|n|} {\mathcal{I}}_{s}(2\pi n
\beta) {\mathcal{K}}_{s}(2\pi n\alpha) e(nx).$$ Inserting this into , we obtain the assertion.

We recall the following lemma from [@Br1]:

\[lHA\] Let $y_1 y_2>A>0$. The function $H^A_s(z_1,z_2)$ defined by (\[HAs\]) has the Fourier expansion $$H^A_s(z_1,z_2) = \sum_{\nu\in\partial_F^{-1}} b_s^A(\nu,y_1,y_2) e(\nu x_1+\nu'x_2)$$ with $$\begin{aligned}
b^A_s(0,y_1,y_2) &= \frac{\pi \Gamma(s-1/2)^2}{ 2\sqrt{p}\,\Gamma(2s)}(4A)^s (y_1 y_2)^{1-s},\\
b^A_s(\nu,y_1,y_2)&=\pi\sqrt{\frac{A}{ p|\nu\nu'|}}\,I_{2s-1}(4\pi\sqrt{A|\nu\nu'|}){\mathcal{K}}_{s}(2\pi |\nu| y_1) {\mathcal{K}}_{s}(2\pi |\nu'| y_2), \quad\nu\nu'>0,\\
b^A_s(\nu,y_1,y_2)&=\pi\sqrt{\frac{A}{ p|\nu\nu'|}}\,J_{2s-1}(4\pi\sqrt{A|\nu\nu'|}){\mathcal{K}}_{s}(2\pi |\nu| y_1)
{\mathcal{K}}_{s}(2\pi |\nu'| y_2), \quad\nu\nu'<0.\end{aligned}$$

The following identity of finite exponential sums is crucial for the main result of the present paper.

\[GaHb\] Let $a\in{\mathbb{N}}$, $m\in{\mathbb{Z}}$, and $\nu\in\partial_F^{-1}$. Then $$\label{eq:key}
\frac{1}{ a } \,\tilde G_a(mp^2,\nu) = \sum_{\substack{r|\nu \\ r|a}} \epsilon_p(r) H_{a/r}\left( \frac{p\nu\nu'}{
r^2},m\right),$$ where the finite exponential sums $\tilde G_a(m,\nu)$ 