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 $n$?

  • 5
    $\begingroup$ 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
  • 1
    $\begingroup$ 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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.