  The Zeta function of a curve $C$ over a finite field may be expressed in
terms of the characteristic polynomial of a unitary matrix $\Theta_C$.
Following the work of Rudnick, we compute the expected value of
$\mbox{tr}(\Theta_C^n)$ over the moduli space of hyperelliptic curves of genus
$g$, over a fixed finite field $\mathbb{F}_q$, in the limit of large genus. As
an application, we compute the expected value of the number of points on $C$ in
$\mathbb{F}_{q^n}$ as the genus tends to infinity. We also look at biases in
both expected values for small values of $n$.
