  Let H(N) denote the Hurwitz class number. It is known that if $p$ is a prime,
then {equation*} \sum_{|r|<2\sqrt p}H(4p-r^2) = 2p. {equation*} In this paper,
we investigate the behavior of this sum with the additional condition $r\equiv
c\pmod m$. Three different methods will be explored for determining the values
of such sums. First, we will count isomorphism classes of elliptic curves over
finite fields. Second, we will express the sums as coefficients of modular
forms. Third, we will manipulate the Eichler-Selberg trace for ula for Hecke
operators to obtain Hurwitz class number relations. The cases $m=2,3$ and 4 are
treated in full. Partial results, as well as several conjectures, are given for
$m=5$ and 7.
