Explicit Evaluation of Double Gauss Sums

We present an explicit evaluation of the double Gauss sum $\displaystyle G(a,b,c;S;p^n):=\sum_{x,y=0}^{p^n-1} e^{2\pi i S(ax^2+bxy+cy^2)/p^n}$,where$a, b, c$areintegerssuchthat$\gcd(a,b,c)=1$,$p$isaprime,$n$ is a positive integer, and $S$ is an integer coprime to $p$.