Abstract
We compute the divisor of the modular equation on the modular curve $\Gamma_0(N) \setminus \mathbb{H}^*$ and then find recurrence relations satisfied by the modular traces of the Hauptmodul for any congruence subgroup $\Gamma_0(N)$ of genus zero. We also introduce the notions and properties of $\Gamma$-equivalence and $\Gamma$-reduced forms about binary quadratic forms. Using these, we can explicitly compute the recurrence relations for $N = 2,3,4,5$.
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