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Let $f\in S^\text{new}_k(\Gamma_0(N))$ be a newform . Are all its Fourier coefficients real? Of course the Hecke operators $T_n$ are selfadjoint for $(n,N)=1$, but is it also true for all $n$?

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    $\begingroup$ Yes. The form $\bar f$ is also a new form with the same Hecke eigenvalues at almost every prime $p$ by your argument, so they are equal by the strong multiplicity one theorem. $\endgroup$
    – Joël
    Jan 9, 2014 at 20:09
  • $\begingroup$ Can yo explain why $\overline{f}$ have the same eigenvalues. I tried to check this, but i get the conjugate eigenvalues. $\endgroup$
    – Abdullah.Y
    Jan 9, 2014 at 22:55
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    $\begingroup$ The reference for this is Proposition 3.2 of Ken Ribet's "Galois Representations Attached to Eigenforms with Nebentypus". $\endgroup$
    – Jeff H
    Jan 9, 2014 at 23:33
  • $\begingroup$ Thanks!The idea with the strong multiplicity one theorem is nice. $\endgroup$
    – Abdullah.Y
    Jan 10, 2014 at 0:39

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