Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
up vote 3 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.