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 ?