Gauss sum variation $\sum_{n=0}^{p-1}\left(\frac{a+bn}{p}\right)\zeta_p^{cn} = ?$

Question

I'm having trouble evaluating this, for $a, b, p$ all pairwise coprime, $p$ an odd prime, $c$ any integer.

$$\sum_{n=0}^{p-1}\left(\frac{a+bn}{p}\right)\zeta_p^{cn}$$

Any help/references would be appreciated! Thanks.

Answer 1

I'll assume $c$ is also coprime to $p$. Then $c\equiv br\pmod p$ for some $r$. The sum is $$\zeta^{-ra}\sum_n\left(\frac{a+bn}p\right)(\zeta^r)^{a+bn} =\zeta^{-ra}\sum_n\left(\frac{m}p\right)(\zeta^r)^m$$ which is a conventional Gauss sum.