  In this paper, we study moments of Hurwitz class numbers associated to
imaginary quadratic orders restricted into fixed arithmetic progressions. In
particular, we fix $t$ in an arithmetic progression $t\equiv m\pmod{M}$ and
consider the ratio of the $2k$-th moment to the zeroeth moment for $H(4n-t^2)$
as one varies $n$. The special case $n=p^r$ yields as a consequence asymptotic
formulas for moments of the trace $t\equiv m\pmod{M}$ of Frobenius on elliptic
curves over finite fields with $p^r$ elements.
