In this talk, we investigate the bias from the first non-vanishing term (on average) after the main term in the asymptotic formula for the second moment of elliptic curves over finite fields whose Trace of Frobenius lies in a certain arithmetic progression. This is a finite field version of a conjecture of Steven J. Miller, who considered the same question for the reduction from a one-parameter family of elliptic curves over the rationals. We investigate this asymptotic formula through moments of certain Hurwitz class numbers. This is joint work with Sudhir Pujahari and Zichen Yang, and builds off of previous work with Kathrin Bringmann and Sudhir Pujahari.