Explicit evaluation of certain exponential sums of binary quadratic functions

Let 0<α1< <αk be integers and , ak≠0. Define where mn and . 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(ax2α+1+cx,n) explicitly for all α>0, , mn and .