Are the eigenvalues of the Hecke operator always real? I am considering the Hecke operators $T_n$ acting on the space $M_k(\text{SL}_2(\mathbb{Z}))$ of weight $k$ modular forms of level 1. Are their eigenvalues always real?
I have read somewhere that the Fourier coefficients of a normalized eigenform are real. The coefficients are precisely the eigenvalues right? Is this correct?
 A: In general, if $\pi$ is an automorphic representation of $\mathrm{GL}(n)$ with contragradient $\tilde\pi$, then $L(s,\tilde\pi)=\bar L(\bar s,\pi)$. In particular, the Dirichlet coefficients of $L(s,\pi)$ are real iff $\tilde\pi\cong\pi$. In the case of $n=2$ we have $\tilde\pi\cong\pi\otimes\omega^{-1}$, where $\omega$ is the central character of $\pi$ (classically the Nebentypus of the forms contained in $\pi$). Hence if $n=2$ and $\omega=1$ as in your case, the Dirichlet coefficients of $L(s,\pi)$, i.e. the Hecke eigenvalues associated with $\pi$, are real.
A: More simply, the eigenvalues are real because the Hecke operators are Hermitian for the Peterson scalar product (a fact which can be checked by a straightforward computation). See for example the introduction by Serre on modular forms in Cours d'arithmétique.
A: The Hecke eigenvalue tells you what principal series of $GL_2(2,\mathbb{Q}_p)$ is associated to the modular froms at the prime $p$. For $\mathrm{SL}_2(\mathbb{Z})$, you have only unramified principal series representation at $p$. You know that they cannot be complementary series representations by Deligne's proof of the Ramanujan conjecture. If I do remember correctly the Eigenvalue of $T_p$ is $p^{-s} + p^{s}$ or something like that with $\Re s = 0$, when $s$ is the parameter of the principal series represenation. So yes!
