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Consider Eisenstein series of weight zero, i.e.

$E_{\mathfrak{a}}(z,\ s,\ \chi) = \sum_{ \gamma \in \Gamma_{\mathfrak{a}} \backslash \Gamma } \bar{\chi}(\gamma) (Im\sigma_{\mathfrak{a}}^{-1} \gamma z)^s$,

where $\chi$ is a multiplier system of weight zero ( $\chi\ :\ \Gamma \rightarrow \mathbb{C}^*$ is a group homomorphism) singular at cusp $\mathfrak{a}$. Then my first question is that why this series converges absolutely in $Re(s)>1$?

My second question is how to calculate the following summation:

$\sum_{d\ (mod c)}\ \epsilon_d(\frac{c}{d})$, where $\gamma =$ $[ \begin{pmatrix} a & b\ c & d \end{pmatrix} ]$ $\in \Gamma_0(4)$, $(\frac{c}{d})$ is the extended quadratic residue symbol and $c = b^2.$

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# The convergence of Eisenstein series of weight zero

Consider Eisenstein series of weight zero, i.e.

$E_{\mathfrak{a}}(z,\ s,\ \chi) = \sum_{ \gamma \in \Gamma_{\mathfrak{a}} \backslash \Gamma } \bar{\chi}(\gamma) (Im\sigma_{\mathfrak{a}}^{-1} \gamma z)^s$,

where $\chi$ is a multiplier system of weight zero ( $\chi\ :\ \Gamma \rightarrow \mathbb{C}^*$ is a group homomorphism) singular at cusp $\mathfrak{a}$. Then my first question is that why this series converges absolutely in $Re(s)>1$?

My second question is how to calculate the following summation:

$\sum_{d\ (mod c)}\ \epsilon_d(\frac{c}{d})$, where $\gamma =$ $[ \begin{pmatrix} a & b\ c & d \end{pmatrix} ]$ $\in \Gamma_0(4)$, and $c = b^2.$