I was wondering whether anyone knows how to approach the following two generalizations of the quadratic Gauss sum:

Given integers r,s with gcd(r,s)=1 and integers a,b,N

$F(r,s,N,a,b) = \sum_{w = 0}^{rsa}(-1)^{b w}(\sin\frac{\pi w}{s}) \exp(\pi i w^2\frac{N}{2 rs}) $

$G(r,s,N,a,b) = \sum_{w = 0}^{rsa}(-1)^{b w}(\sin\frac{\pi w}{r})(\sin\frac{\pi w}{s}) \exp(\pi i w^2\frac{N}{2 rs}) $

Note that removing the sine terms and the sign, setting a = 2 and N = 2 gives the classical quadratic Gauss sum.

Some experimentation suggests that

$F(r,s,N,a,b) = 0$ for all integers b,N, r,s if a is even and (r,s) =1 and

$G(r,s,N,a,b) = 0$ for all a,b,N and r,s with (r,s) =1

Is there a good reason for these sums to vanish? Or a clean proof/reference?

Is it possible to evaluate F in the case a = 1? It seems to be non-zero then.

I tried reducing to the original Gauss sum by completing the square but this seems to get quite ugly.

More generally, do such Gauss-like sums have a more natural generalization that turns up somewhere?

Thanks