 Solution

If $\gamma \in \Gamma (N)$, then $(x,y)\gamma \equiv (x,y)(\mathrm{mod}N)$, so part c gives invariance. It remains to check the cusps. For any $\sigma \in S{L}_2(\mathbb{Z})$,As $\mathrm{Im}\tau \to \infty$, the terms with $c=0$ give a finite constant and the locally uniform estimate for the terms with $c\ne 0$ gives boundedness. A periodic holomorphic function bounded at infinity has a Fourier expansion with no negative powers. Thus every slash transform is holomorphic at infinity, which is holomorphy at every cusp. Hence $G_k^{(x,y)}$ is the congruence-class Eisenstein series in $M_k(\Gamma (N))$.