Determination of the Fricke families

Abstract

For a positive integer $N$ divisible by $4$, let ${\mathcal{O}}_N^1(\mathbb{Q})$ be the ring of weakly holomorphic modular functions for the congruence subgroup ${\Gamma }^1(N)$ with rational Fourier coefficients. We present explicit generators of the ring ${\mathcal{O}}_N^1(\mathbb{Q})$ over $\mathbb{Q}$, from which we are able to classify all Fricke families of such level $N$.