I was wondering if there exists a Poisson Summation formula (like the one existing with primitive character) for imprimitive Dirichlet characters ?

For a primitive Dirichlet character $\chi$ (providing good properties on f and its Fourier transform $\hat{f}$) we have:

$ \sum\limits_{n=-\infty}^{\infty}\chi(n) f(\frac{n}{q}x) =\frac{K}{x} \sum\limits_{n=-\infty}^{\infty} \overline{\chi(n)} \hat{f}(\frac{n}{x})$

But for imprimitive characters ?

I was wondering if there exists a Poisson Summation formula (like the one existing with primitive character) for imprimitive Dirichlet characters ?

For a primitive Dirichlet character $\chi$ we have:

$$ \sum\limits_{n=-\infty}^{\infty}\chi(n) f\bigg(\frac{n}{q}x\bigg) =\frac{K}{x} \sum\limits_{n=-\infty}^{\infty} \overline{\chi(n)} \hat{f}\bigg(\frac{n}{x}\bigg)$$

But for imprimitive characters ?

I was wondering if there exists a Poisson Summation formula (like the one existing with primitive character) for imprimitive Dirichlet characters ?

I was wondering if there exists a Poisson Summation formula (like the one existing with primitive character) for imprimitive Dirichlet characters ?

For a primitive Dirichlet character $\chi$ (providing good properties on f and its Fourier transform $\hat{f}$) we have:

$ \sum\limits_{n=-\infty}^{\infty}\chi(n) f(\frac{n}{q}x) =\frac{K}{x} \sum\limits_{n=-\infty}^{\infty} \overline{\chi(n)} \hat{f}(\frac{n}{x})$

But for imprimitive characters ?