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In this paper, we study moments of Hurwitz class numbers associated toimaginary quadratic orders restricted into fixed arithmetic progressions. Inparticular, we fix $t$ in an arithmetic progression $t\equiv m\pmod{M}$ andconsider 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 asymptoticformulas for moments of the trace $t\equiv m\pmod{M}$ of Frobenius on ellipticcurves over finite fields with $p^r$ elements.

Distribution of moments of Hurwitz class numbers in arithmetic progressions and holomorphic projection