Recurrence relations satisfied by the traces of singular moduli for ${\mathrm{\Gamma }}_0(N)$

We compute the divisor of the modular equation on the modular curve ${\mathrm{\Gamma }}_0(N)\mathrm{\backslash }{\mathbb{H}}^{\ast }$ and then find recurrence relations satisfied by the modular traces of the Hauptmodul for any congruence subgroup ${\mathrm{\Gamma }}_0(N)$ of genus zero. We also introduce the notions and properties of $\mathrm{\Gamma }$-equivalence and $\mathrm{\Gamma }$-reduced forms about binary quadratic forms. Using these, we can explicitly compute the recurrence relations for $N = 2, 3, 4, 5$.