Voronoi's summation formula says something like $$\sum _{n}d(n)e(an/q)w(n)=\sum _nd(n)e(\overline an/q)e(\sqrt {nx}/q)\hat w(n)$$
where $w(n)$ is a smooth weight function and $\hat w$ is some integral transform of $w$.
I would like a similar formula but with a Dirichlet character (of general modulus) in the sum as well. But when I google around, nothing seems to come up. My guess is, that my question doesn't really make sense. Either it's not really possible, or it's a weird specificaiton. I find lots of generalisations of the above formula but I don't know how to think of the general (Hecke/Mass/modular/similar-adjectives) coefficients that come up in that case - perhaps they contain Dirichlet characters.
Anyway, anyone who can sense what I'm missing and can give a few comments would be great.
P.S. I'm aware I could write the character as a sum over an exponential, but this leads to the RHS above being something like $\Sigma _b^r\chi (b)e(n(\overline a/q-b/r))$ and I'm not sure if I can do anything with this.