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 non-eigen) 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 $c_{2} = -\frac{1}{24 \sqrt{144169}}$.

| cite | improve this answer | |
  • 1
    $\begingroup$ A charming, decisive, minimalist example! :) $\endgroup$ – paul garrett Jul 3 at 19:29

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.