# Are the Fourier coefficients of a new form real

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 $n$?

• 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.
– Joël
Jan 9 '14 at 20:09
• Can yo explain why $\overline{f}$ have the same eigenvalues. I tried to check this, but i get the conjugate eigenvalues. Jan 9 '14 at 22:55
• The reference for this is Proposition 3.2 of Ken Ribet's "Galois Representations Attached to Eigenforms with Nebentypus". Jan 9 '14 at 23:33
• Thanks!The idea with the strong multiplicity one theorem is nice. Jan 10 '14 at 0:39