Are the eigenvalues of the Hecke operators $T_n$ for $M_k(\text{SL}_2(\mathbb{Z}))$ always real? I think I have an answer but I am not confident with my arguments.
If $f$ is a normalized eigenform, then $f$ have real Fourier coefficients. And if $f$ is a normalized eigenform, then its coefficients are precisely the eigenvalues of $T_n$. Thus $T_n$ must have real eigenvalues. Is this correct?