We establish a simple inductive formula for the trace Trknew(Γ0(8),p) of the pth Hecke operator on the space Sknew(Γ0(8)) of newforms of level 8 and weight k in terms of the values of 3F2-hypergeometric functions over the finite field Fp. Using this formula when k = 6, we prove a conjecture of Koike relating Tr6new(Γ0(8),p) to the values 6F5(1)p and 4F3(1)p. Furthermore, we find new congruences between Trknew(Γ0(8),p) and generalized Apéry numbers.
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Frechette et al. (2004) studied this question.