  The zeta function of a curve $C$ over a finite field may be expressed in
terms of the characteristic polynomial of a unitary matrix $\Theta_C$.
  We develop and present a new technique to compute the expected value of
$\mathrm{Tr}(\Theta_C^n)$ for various moduli spaces of curves of genus $g$ over
a fixed finite field in the limit as $g$ is large, generalizing and extending
the work of Rudnick and Chinis.
  This is achieved by using function field zeta functions, explicit formulae,
and the densities of prime polynomials with prescribed ramification types at
certain places as given by Bucur, David, Feigon, Kaplan, Lal\'in and Wood
[BDF$^+$16] and by Zhao.
  We extend [BDF$^+$16] by describing explicit dependence on the place and give
an explicit proof of the Lindel\"{o}f bound for function field Dirichlet
$L$-functions $L(1/2 + it, \chi)$.
  As applications, we compute the one-level density for hyperelliptic curves,
cyclic $\ell$-covers, and cubic non-Galois covers.
