Distribution and divisibility of the Fourier coefficients of certain Hauptmoduln.

Suppose j N (τ) and j N ⁎ (τ) are the Hauptmoduln of the congruence subgroup Γ 0 (N) and the Fricke group Γ 0 ⁎ (N) , respectively. In [7] , the authors predicted that, like Klein's j -function, the Fourier coefficients of j N (τ) and j N ⁎ (τ) in some arithmetic progression are both even and odd with density 1 2. In this article, we can find some arithmetic progression of n where the Fourier coefficients of j 6 (τ) (resp. j 6 ⁎ (τ) and j 10 (τ)) are almost always even. Furthermore, using Hecke eigenforms and Rogers-Ramanujan continued fraction, we obtain infinite families of congruences for j 6 (τ) , j 6 ⁎ (τ) , j 10 (τ) , and j 10 ⁎ (τ). [ABSTRACT FROM AUTHOR]