Fourier coefficients of noncongruence cuspforms

Given a finite index subgroup of $SL_2(\mathbb Z)$ 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.

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