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

For $\delta\in\Gamma(1)$, right multiplication by $\delta$ permutes $\Gamma_\infty\backslash\Gamma(1)$. The <automorphy factor> identity
$$
j(\gamma\delta,\tau)
=j(\gamma,\delta\tau)j(\delta,\tau)
$$
therefore gives
$$
E_{k,s}(\delta\tau)
=j(\delta,\tau)^kE_{k,s}(\tau).
$$
The <modular form> $f$ obeys the same weight-$k$ transformation law, while
$$
\operatorname{Im}(\delta\tau)^k
=\frac{\operatorname{Im}(\tau)^k}{|j(\delta,\tau)|^{2k}}.
$$
Consequently
$$
\begin{aligned}
&f(\delta\tau)\overline{E_{k,s}(\delta\tau)}
\operatorname{Im}(\delta\tau)^k\\
&\quad=
j(\delta,\tau)^k\overline{j(\delta,\tau)}^k
|j(\delta,\tau)|^{-2k}
f(\tau)\overline{E_{k,s}(\tau)}\operatorname{Im}(\tau)^k\\
&\quad=
f(\tau)\overline{E_{k,s}(\tau)}\operatorname{Im}(\tau)^k.
\end{aligned}
$$
Thus the product is invariant under the weight-zero action of $\Gamma(1)$, as described by the <invariant product with a weight-k real-analytic Eisenstein series>.

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 137 / 4 / b / Solution

{title2=$j(\gamma,\tau)=c\tau+d$}

For $\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, the automorphy factor is $j(\gamma,\tau)=c\tau+d$. It obeys the cocycle identity
$$
j(\gamma\delta,\tau)=j(\gamma,\delta\tau)j(\delta,\tau).
$$


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