Fourier coefficients of noncongruence cuspforms

Abstract

Given a finite index subgroup of $S{L}_2(\mathbb{Z})$ with modular curve defined over $\mathbb{Q}$, under the assumption that the space of weight $k$ ($\ge 2$) cusp forms is $1$-dimensional, we show that a form in this space with Fourier coefficients in $\mathbb{Q}$ has bounded denominators if and only if it is a congruence modular form.