Over at mathoverflow there is a question regarding the twisted Poisson summation formula:
$$ \sum_{n \in \mathbb{Z}} \chi(n) f \left( \frac{nx}{q} \right) = \frac{K}{x} \sum_{n \in \mathbb{Z}} \overline{\chi}(n) \hat{f} \left( \frac{n}{x} \right) $$
where $\chi $ is a primitive Dirichlet character modulo $q$ and $K$ is the root number of $\chi$.
The question concerns whether this formula extends to non-primitive characters. As a preface to one of the answers @Myshkin writes
"The reason we can get that (twisted) Poisson summation formula in the first place is that in the primitive case you can interpolate the character to a smooth real function via Gauss sums."
I'm assuming here that this refers to using the discrete fourier transform of $\chi$ to write:
$$ \chi(n) = \frac{G(n,\overline{\chi})}{G(1,\overline{\chi})} = \frac{\sum_{a = 1}^q \overline{\chi} (a) e^{\frac{2 \pi i a n}{q}}}{\sum_{a = 1}^q \overline{\chi} (a) e^{\frac{2 \pi i a}{q}}} $$
which holds for primitive characters and is continuous in $n$.
However I can't find a proof anywhere that uses the analytic properties of this expression, which seems to be implied in the comment. Does anybody have a reference for such a proof?
The proofs I have seen use a version of the summation formula that holds for all functions $c:\mathbb{Z}/ q \mathbb{Z} \rightarrow \mathbb{C}$ of which $\chi$ is just a particularly neat example.
I'm less concerned with the summation formula itself but curious as to whether this 'interpolated' Dirichlet character has any interesting properties other than being a DFT. As such I'd also be interested in other proofs that make use of such properties.