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DOI: 10.1016/j.jnt.2023.01.014

# Distribution and divisibility of the Fourier coefficients of certain Hauptmoduln

Mar 2023

Suppose $j_N(\tau)$ and $j_N^{*}(\tau)$ are the Hauptmoduln of the congruencesubgroup $\Gamma_0(N)$ and the Fricke group $\Gamma^{*}_0(N)$, respectively. In[7], the authors predicted that, like Klein's $j$-function, the Fouriercoefficients of $j_N(\tau)$ and $j_{N}^{*}(\tau)$ in some arithmeticprogression are both even and odd with density $\frac{1}{2}$. In this article,we can find some arithmetic progression of $n$ where the Fourier coefficientsof $j_6(\tau)$ (resp. $j_6^{*}(\tau)$ and $j_{10}(\tau)$) are almost alwayseven. Furthermore, using Hecke eigenforms and Rogers-Ramanujan continuedfraction, we obtain infinite families of congruences for $j_6(\tau)$,$j_6^{*}(\tau)$, $j_{10}(\tau),$ and $j_{10}^{*}(\tau)$.

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