Let $S\_k(\Gamma\_1(N))(\mathbb{Z})$ be the set of modular forms of weight $k$ and level $N$ with integer Fourier coefficients. Then is true that any cusp form can be written as $\mathbb{Q}$ linear combination of Hecke eigenforms with integer coefficients ?



Cusp forms with integer Fourier-coefficients
No. Take $k = 24$ and $N = 1$. Then $\Delta^{2} = q^{2} - 48q^{3} + 1080q^{4} + \cdots \in S\_{K}(\Gamma\_{1}(N),\mathbb{Z})$. However, if we write $\Delta^{2} = c\_{1} f\_{1} + c\_{2} f\_{2}$, where $f\_{1}$ and $f\_{2}$ are the Hecke eigenforms (with coefficients in $\mathbb{Q}(\sqrt{144169})$) then (if we order $f\_{1}$ and $f\_{2}$ appropriately), $c\_{1} = \frac{1}{24 \sqrt{144169}}$ and $c\_{2} = -\frac{1}{24 \sqrt{144169}}$.


