The modular curve $Y(n)$ which over $\mathbb{C}$ is $\mathcal{H}/\Gamma(n)$ is often viewed as a curve defined over $\mathbb{Q}(\zeta_n)$. However, if one twists the moduli problem to be given by pairs $(E,\alpha)$ where $E$ is an elliptic curve over some $\mathbb{Q}$-scheme $S$, and $\alpha$ is an isomorphism: $$\alpha : E[n]\stackrel{\sim}{\longrightarrow}\mathbb{Z}/n\times\mu_n$$ "of determinant 1", then one obtains a model of $Y(n)$ over $\mathbb{Q}$.

The usual $q$-expansion arguments show that $Y(n)_{\mathbb{Q}(\zeta_n)}$ is isomorphic to the Spec of the ring of modular functions (modular forms of weight 0) for $\Gamma$ whose Fourier coefficients lie in $\mathbb{Q}(\zeta_n)$.

My question is: Is there an $n\ge 3$ such that $Y(n)_\mathbb{Q}$ isomorphic to the Spec of modular functions for $\Gamma$ whose Fourier coefficients lie in $\mathbb{Q}$? (ie, are there enough of these functions?)

My guess is that the answer is no, but how would one argue this?