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 ?
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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 non-eigen) form with integer coefficients. $\endgroup$– KimballJul 3, 2020 at 19:28
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$\begingroup$ Related: mathoverflow.net/questions/364787/… $\endgroup$– David LoefflerJul 4, 2020 at 8:46
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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 $c_{2} = -\frac{1}{24 \sqrt{144169}}$.
answered Jul 3, 2020 at 19:25
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1$\begingroup$ A charming, decisive, minimalist example! :) $\endgroup$ Jul 3, 2020 at 19:29