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This paper gives a quick way to construct all modular functions for the group$\Gamma_0(N)$ having only a pole at $\tau = i \infty$. We assume that we aregiven two modular functions $f,g$ for $\Gamma_0(N)$ with poles only at $i\infty$ and coprime pole orders. As an application we obtain two new identitiesfrom which one can derive that $p(11n+6)\equiv 0\pmod{11}$, here $p(n)$ is theusual partition function.

Computing an order complete basis for $M^{\infty}(N)$ and Applications