Explicit evaluation of certain exponential sums of binary quadratic functions

Let 0<α1<⋯<αk be integers and , ak≠0. Define S(f,n)=∑x∈Fn2e(f(x)) where m|n and e(x)=(−1)TrFn2/F2(x). We establish a relation among S(f,n) for all n with the same 2-adic order. When ν2(α1)=⋯=ν2(αk), where ν2 is the 2-adic order function, we are able to compute S(f,n) explicitly for all n with a given f. Moreover, we are able to compute S(axα2+1+cx,n) explicitly for all α>0, a∈Fm2, m|n and c∈Fn2.