Are there infinitely many (linearly independent) cuspidal eigenforms for $\Gamma(1)$ with integer coefficients?


Someone told me that the Hecke algebra is conjectured to act irreducibly on the space of modular forms of level 1, so there would be no eigenforms if $\mathrm{dim} S\_k > 1$, i.e. for $k \geq 12$.



modular eigenforms with integral coefficients [Maeda's Conjecture]
I will replace  *modular form* by *cuspform* in your question, just to avoid the trivial answer of *yes*, because of the Eisenstein series. Given this, numerical evidence suggests that the eigenforms of weight $k$ are all conjugate (in the sense that their $q$-expansions are all algebraically conjugate under the action of $Gal(\overline{\mathbb Q}/{\mathbb Q})$). If this were the case, then the answer to your question would be *no*. Since this question is open, I'm pretty sure the answer to your question is not known.


There is a lot of evidence for this conjugacy statement, but the only suggestion I know for why it's true is essentially that it's the simplest possibility, given that no reason is known for it to be false (in contrast to the case of say fixing the weight to be 2, but letting the
level grow; in that case elliptic curves give rise to integral eigenforms, and there
are elliptic curve of arbitrarily high conductor). 


