Linear independence of primitive Dirichlet characters and convolution

Question

This is not an exercise but merely a question I have. Fix $N \in \mathbb{N}$ and suppose there exist some values $a_k \in \mathbb{C}$, for $k \in \mathbb{Z}_N$, such that $$ \sum_{k \in \mathbb{Z}_N} a_k \chi(m - k) = 0 $$ for every $m \in \mathbb{Z}_N$ and every primitive Dirichlet character $\chi$ modulo $N$. Prove or disprove that each $a_k = 0$ for every $k \in (\mathbb{Z}/N\mathbb{Z})^\times$. Thanks!