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 $M\_k(1)$. 



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> Does there exist a basis in $\mathbb{Z}[[q]]$ for $M\_k(N)$?
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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 $M\_k(N) \cap \mathbb{Z}[[q]]$. 



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> 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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Is there a Miller basis for M_k(N)?
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).


