Recurrence relations satisfied by the traces of singular moduli for Γ₀ (n)

We compute the divisor of the modular equation on the modular curve Γ₀ (N) \ ℍ^∗ and then find recurrence relations satisfied by the modular traces of the Hauptmodul for any congruence subgroup Γ₀ (N) of genus zero. We also intro-duce the notions and properties of Γ-equivalence and Γ-reduced forms about binary quadratic forms. Using these, we can explicitly compute the recurrence relations for N = 2, 3, 4, 5.