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 ?

3$\begingroup$ No, very few eigenforms can be scaled to have integer FCs. However, given any eigenform scaled so its coefficients are algebraic integers, you can take the "trace" by summing up all conjugate forms to get a (typically noneigen) form with integer coefficients. $\endgroup$ – Kimball Jul 3 at 19:28

$\begingroup$ Related: mathoverflow.net/questions/364787/… $\endgroup$ – David Loeffler Jul 4 at 8:46
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 $c_{2} = \frac{1}{24 \sqrt{144169}}$.

1$\begingroup$ A charming, decisive, minimalist example! :) $\endgroup$ – paul garrett Jul 3 at 19:29