  We consider the expected value of $\mbox{Tr}(\Theta_C^n)$ where $C$ runs over
a thin family of $r$-cyclic covers of $\mathbb{P}^1_{\mathbb{F}_q}$ for any
$r$. We obtain many lower order terms dependent on the divisors of $r$. We use
these results to calculate the one-level density of the family and hypothesize
a refined one-level density result.
