In this paper, we relax the condition in Theorem 2 of  to include a larger class of $N$. We then prove similar results on congruence subgroups other than $\Gamma_0(N)$.
"A Class of Modular Functions." Funct. Approx. Comment. Math. 38 (2) 227 - 232, September 2008. https://doi.org/10.7169/facm/1229696541