Let $e(x):=e^{2\pi i x}$. Put $G(N):=\sum_{n\pmod{N}} e(\frac{n^2}{N})$.
I wonder how to prove directly (without invoking Dirichlet's formula) the following two elementary identities from Iwaniec's Topics in Automorphic forms, page 5 (he says the proof is elementary).
$G(N^2)=N, G(N^3)=NG(N).$
There might be some tricks here that I am not familiar with. Any hints or solutions will be appreciated!