Is there a connection between the values of a function on arithmetic progressions and the values of its DFT also on arithmetic progressions?

Question

Let $f : \mathbb{Z}_N \to \mathbb{C}$. Its DFT is given by $$ \widehat{f}(m) = \sum_{j \in \mathbb{Z}_N}f(j) \zeta_N^{mj}, $$ where $\zeta_N$ is a primitive $N$-th root of unity. Let $a,r$ be integers. Is there a connection between the amount of zero values in the sequences $$ (f(m))_{m \in \mathbb{Z}_N, \ m \equiv a \pmod{r}} $$ and $$ (\widehat{f}(m))_{m \in \mathbb{Z}_N, \ m \equiv a \pmod{r}}? $$ Basically I wonder if one can switch from counting the number of zeros of $f$ in some arithmetic progression to the number of zeros of $\widehat{f}$ in some (maybe not the same) arithmetic progression?