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

Abstract

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 are given 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 identities from which one can derive that $p(11n+6)\equiv 0(\mathrm{mod}11)$, here $p(n)$ is the usual partition function.

6 pages