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The space $M_k(1)$ of modular forms of level $1$ has a unique basis $\left\{ f_0,f_1,\ldots,f_d\right\}$ such that $a_i(f_j) = \delta_{ij}$ for all $0\leq i,j \leq d$, and $f_i \in \mathbb{Z}[[q]]$ for all $0 \leq i \leq d$. This is often called the Miller basis for $M_k(1)$.

Does there exist a basis in $\mathbb{Z}[[q]]$ for $M_k(N)$?

This would seem quite incredible. Maybe the following analogon of the Miller basis might be true in higher level. Let us denote $M_k(N,\mathbb{Z})$ the elements in $M_k(N) \cap \mathbb{Z}[[q]]$.

Is $M_k(N,\mathbb{Z})$ a free $\mathbb{Z}$-module? If so, what is its rank in function of $N,k$?

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up vote 3 down vote accepted

The space $M_k(N)$ has a basis in $\mathbb{Z}[[q]]$ for any $N$ and $k$. This is a straightforward consequence of Eichler-Shimura theory and will be in any decent textbook on modular forms (e.g. Diamond + Shurman, Miyake, Lang).

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