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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!

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up vote 1 down vote accepted

Write $\alpha = c\gamma + \beta$, where $c$ lies in $\mathbb{Z}[i]$. Now the exponential term looks like $e(z|c \gamma + \beta|^2)$, where $c$ is the variable. Rewrite it as $e((z |\gamma|^2) |c + \beta/\gamma|^2)$. In your formula, use $c$ as $\alpha$, $\beta/\gamma$ as $\beta$ and $z |\gamma|^2$ as $z$.

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