  We study the moments of $\mbox{Tr}(\Theta_\chi)$ as $\chi$ runs over
Dirichlet characters defined over $\mathbb{F}_q[T]$ of fixed order $r$. In
particular, we show that after an appropriate normalization, the $q$-limit of
the power sum moments behave like the power sum moments of the group of unitary
matrices multiplied by a weight function.
