Distribution and divisibility of the Fourier coefficients of certain Hauptmoduln

Abstract

Suppose $j_N(\ddot{I}„)$ and $j_N^{\ast }(\ddot{I}„)$ are the Hauptmoduln of the congruence subgroup $\hat{I}{“}_0(N)$ and the Fricke group $\hat{I}{“}_0^{\ast }(N)$, respectively. In [7], the authors predicted that, like Klein's $j$-function, the Fourier coefficients of $j_N(\ddot{I}„)$ and $j_N^{\ast }(\ddot{I}„)$ in some arithmetic progression are both even and odd with density $\frac{1}{2}$. In this article, we can find some arithmetic progression of $n$ where the Fourier coefficients of $j_6(\ddot{I}„)$ (resp. $j_6^{\ast }(\ddot{I}„)$ and $j_{10}(\ddot{I}„)$) are almost always even. Furthermore, using Hecke eigenforms and Rogers-Ramanujan continued fraction, we obtain infinite families of congruences for $j_6(\ddot{I}„)$, $j_6^{\ast }(\ddot{I}„)$, $j_{10}(\ddot{I}„),$ and $j_{10}^{\ast }(\ddot{I}„)$.

The comments are welcome!