Eigenvalues of Hecke operators are algebraic integers

Question

I want to show that if a normalised modular form of level $1$, say $f$, is a simultaneous eigenform of the Hecke operators $T(n)$ for all $n$, then the corresponding eigenvalues are all algebraic integers. My approach goes like this:

Let $M_k(SL_2(\mathbb{Z}))$ be the space of modular forms of level $1$ and weight $k$. Let $M_k(SL_2(\mathbb{Z}))_{\mathbb{Z}}$ be the subset of those modular forms all of whose Fourier coefficients are integers. Then, one can check that the action of $T(n)$ descends to an action on $M_k(SL_2(\mathbb{Z}))_{\mathbb{Z}}$. Hence, if we can find a $\mathbb{Z}$-basis for $M_k(SL_2(\mathbb{Z}))_{\mathbb{Z}}$ which is also a $\mathbb{C}$-basis for $M_k(SL_2(\mathbb{Z}))$, then, the matrix of the action of $T(n)$ will have integer entries, and so, the eigenvalues will be algebraic integers. So, I only need to find such a basis.

I feel that some combinations of $E_4$ and $E_6$ (the normalised Weierstrass forms) should work, but I haven't been able to work out the details.

Answer 1

You are correct. For a given $k\geqslant 12$ a basis for $M_k$ is given by $E_4^aE_6^b$ with $4a+6b=k$.