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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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3 Answers

up vote 11 down vote accepted

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! :) –  Victor Aricheta Jul 8 '12 at 9:25
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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 $(\mu_2^{-1}, \mu_1^{-1})$. –  plusepsilon.de 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:\ –  plusepsilon.de Jul 8 '12 at 13:49
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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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