Take $q,n\in \mathbb N$. I want to evaluate exactly \[ S_q(n):=\sum _{\chi }\tau (\chi )\chi (n)\] where the sum extends over primitive characters modulo $q$ and where $\tau (\chi )$ denotes the Gauss sum \[ \tau (\chi ):=\sum _{a=1}^q\chi (a)e\left (a/q\right ).\] I guess I would like the $n$ dependence "separated", whatever this might mean.
If I open up the Gauss sum then \[ S_q(n)=\sum _{a=1\atop {(a,q)=1}}^qe\left (a/q\right )\sum _{\chi }\chi (an)\] but I don't know if I can say anything sensible about the inner sum. All I can think of is writing it down explicitly (in terms of indices and primitive roots), which I think means writing it as \[ \sum _{h=1\atop {(h,q)=1}}^qe\left (hv(an)/q\right )=c_q\left (v(an)\right )\] so that \[ S_q(n)=\sum _{a=1\atop {(a,q)=1}}^qe\left (a/q\right )c_q\left (v(an)\right )\] but I still don't see how I can write the $n$ dependence in any useful way.
Am I missing something or is it just not possible?
Small note: Ultimately I will be summing over \[ \sum _{q|Q}f(q)\] so perhaps it's only possible to get a sensible formula after summing over all $q$. But $f$ doesn't have a factor $\chi _0(n)$ (mod $Q$) so I wouldn't know how to combine the primitive characters altogether.