# 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?

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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.

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Thanks! I got it now! :) – Flip Jul 8 '12 at 9:25

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.

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nice answer. One small nitpick: the contragedient of the prinicpal series of $(\mu_1, \mu_2)$ is the p.s. of $(\mu_2, \mu_1)$, not $(\mu_2^{-1}, \mu_1^{-1})$. – Marc Palm Jul 8 '12 at 9:36
Marc: I disagree. The principal series of $(\mu_1,\mu_2)$ and $(\mu_2,\mu_1)$ are isomorphic, while their contragradients are isomorphic to $(\mu_1^{-1},\mu_2^{-1})$. In general, when you induce from $n$ characters of $\mathrm{GL}(1)$ to obtain a representation of $\mathrm{GL}(n)$, you can permute the characters without changing the induced guy. Think also about this: the contragradient's central character is inverse to the original central character. See also Theorem 3.3.5 on Page 305 and (5.21) on Page 332 in Bump: Automorphic forms and representations. – GH from MO Jul 8 '12 at 13:30
Sure thing, was thinking about iso, not contragedient. Sorry:\ – Marc Palm Jul 8 '12 at 13:49

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!

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