# Are the eigenvalues of the Hecke operators always real?

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?

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Yes, in this context (and more generally for newforms on $\Gamma_0(N)$) the eigenvalues of the $T_n$ must be real. What your argument shows is that this is*equivalent* to the statement that a normalized eigenform has real Fourier coefficients. If you accept that the Fourier expansion is necessarily real, you are done.
On the other hand, I would normally prove that the Fourier expansion is real by first proving that the Hecke eigenvalues are real. For this, I would use the fact that $T_n$ is self-adjoint for the Petersson inner product, and that the eigenvalues of a self-adjoint operator are necessarily real.