I have been reading the book Topics in classical automorphic forms for a while, where I encountered a formula in the middle of Sec 8.6 (p143)
$F(z;\beta,\gamma)=\frac {i\gamma}{4|\gamma|^4z^2}\sum_{\alpha\in\mathbb{Z}[i]}\alpha e\biggl(\frac{-|\alpha|^2}{4z|\gamma|^2}+\textrm{Re }\frac{\bar{\alpha}\beta}{\gamma}\biggr),~~~~~~(1)$ where $F(z;\beta,\gamma)=\sum_{\alpha\in\mathbb{Z}[i]\atop \alpha\equiv\beta~(\bmod \gamma)}\alpha e(z|\alpha|^2)~~~~~~(2)$ for any $z\in\mathbb{C}$ with $\textrm{Im }z>0$, $\beta,\gamma\in\mathbb{Z}[i]$ with $\gamma\not=0$.
What we have already known is that, by standard method,
$\sum_{\alpha\in\mathbb{Z}[i]}(\alpha+\beta)e(z|\alpha+\beta|^2)=\frac {i}{4z^2}\sum_{\alpha\in\mathbb{Z}[i]}\alpha e\biggl(\frac{-|\alpha|^2}{4z}+\textrm{Re }\bar{\alpha}\beta\biggr).~~~~~~(3)$
My problem is that how am I supposed to deal with the congruent condition in the subscription of the summation in $(2)$. Sorry if I've asked something trivial. And thanks for any help!