Write $\gamma=\begin{pmatrix}a&b\\r&s\end{pmatrix}$. The <automorphy factor> identity gives
$$
(r\tau+s)^{-k}(c\gamma\tau+d)^{-k}
=((c,d)\gamma(\tau,1)^T)^{-k}.
$$
Right multiplication by $\gamma$ bijects $\mathbb Z^2\setminus\{0\}$ and carries the congruence class $(x,y)$ to $(x,y)\gamma$. Reindexing the absolutely convergent series yields
$$
G_k^{(x,y)}|_k\gamma=G_k^{(x,y)\gamma}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 137 / 1 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-137/1/c/solution