Are the coefficients of this eigenform totally real?

Question

Let $f \in S_2(\Gamma_0(N))$ (cusp forms) be a normalized Hecke eigenform, and let $K_f$ be the number field obtained by adjoining all its Fourier coefficients to $\mathbb{Q}$.

Then is $K_f$ totally real?

Answer 1

Not necessarily, but yes if f is new.

If f is new then it is uniquely determined by its Hecke eigenvalues away from N. These are eigenvalues of a selfadjoint operator, so they are real.

For old eigenforms it is false; e.g if f is a new eigenform of level N, and p is a prime not dividing N, there are two eigenforms in the oldspace at level Np corresponding to f and neither of them have real Up eigenvalue.