Distribution and divisibility of the Fourier coefficients of certain Hauptmoduln

Suppose $j_N(\tau )$ and $j_N^{\ast }(\tau )$ are the Hauptmoduln of the congruence subgroup ${\mathrm{\Gamma }}_0(N)$ and the Fricke group ${\mathrm{\Gamma }}_0^{\ast }(N)$, respectively. In [7], the authors predicted that, like Klein’s $j$-function, the Fourier coefficients of $j_N(\tau )$ and $j_N^{\ast }(\tau )$ 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 $j_6(\tau )$ (resp. $j_6^{\ast }(\tau )$ and $j_{10}(\tau )$) are almost always even. Furthermore, using Hecke eigenforms and Rogers-Ramanujan continued fraction, we obtain infinite families of congruences for $j_6(\tau )$, $j_6^{\ast }(\tau )$, $j_{10}(\tau ),$ and $j_{10}^{\ast }(\tau )$.