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We consider the expected value of $\mbox{Tr}(\Theta_C^n)$ where $C$ runs overa 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 usethese results to calculate the one-level density of the family and hypothesizea refined one-level density result.

Lower Order Terms for Expected Value of Traces of Frobenius of a Family of Cyclic Covers of $\mathbb{P}^1_{\mathbb{F}_q}$ and One-Level Densities