Let the quadratic Gauss sum be defined by

$\displaystyle G(m;n)=\underset{r=1}{\overset{n}{\sum}}\omega^{mr ^{2}}$, where $\displaystyle \omega=e^{2\pi i/n}$.

I have managed to show that $\displaystyle \omega^{mr^{2}}=\omega^{ms^{2}}$, whenever $\displaystyle r\equiv s\mod\: n)$.

My problem is that I don't know how to deduce that $\displaystyle G(m;n)=\{\sum}\omega^{mr^{2}}$, where the summation extends over ANY complete set of residues.

Could anybody help me with this deduction please?