Real character modular forms: Fourier coefficient real? Let $f$ be a modular form of level $N$ and real character $\chi$ of mod $N$ and weight $k$.
Does the Fourier coefficient or hecke-eigenvalue of $f$ have to be real?
What I knew is that if $N=1$ and $\chi$ trivial then fourier coefficents are real.
 A: I was encouraged to turn my comment into an answer. Theorem 6.20 of Iwaniec's book "Topics in Classical Automorphic Forms" states that if $f(z) = \sum \lambda(n) q^{n}$ is a newform (a normalized Hecke eigenform that is in the new subspace), then
$$ \lambda(n) = \chi(n) \overline{\lambda(n)} $$
and so $\lambda(n)$ is real if $\chi(n) = 1$, while if $\chi(n) \ne 1$ then either $\lambda(n)$ is not real, or $\lambda(n) = 0$.
For example, the space of cusp forms of weight $4$ for $\Gamma_{0}(8)$ with character $\chi_{2}(n) = \left( \frac{2}{n}\right)$ has dimension $2$ and one of the newforms is
$$
  q + (-1 - \sqrt{-7}) q^{2} + 2 \sqrt{-7} q^{3} + (-6 + 2 \sqrt{-7}) q^{4} - 4 \sqrt{-7} q^{5} + \cdots
$$
and the other is the complex conjugate.
It can happen that $\lambda(n)$ is real for all $n$ even if $\chi \ne 1$. For example,
$$
  f(z) = q \prod_{n=1}^{\infty} (1-q^{n})^{3} (1-q^{7n})^{3}
$$
is a newform of weight $3$, level $7$ with character $\chi = \left(\frac{n}{7}\right)$. Such forms must be "CM forms" and arise from a Hecke grossencharacter. (See Theorem 12.5 of Iwaniec's book for details.)
A: When acting on forms with trivial character, the Hecke operators $T_n$ for $n$ coprime to $N$ are Hermitian, so their eigenvalues are always real. This doesn't work for the Hecke operators of index not coprime to $N$; but if you have an eigenform $f$ which is new of level $N$, then its Hecke eigenvalues are all real (since the ones for $p \nmid N$ determine the rest, by the strong multiplicity one theorem). In particular, if $f$ is normalized, then its Fourier coefficients are real too (because they equal the Hecke eigenvalues).
For an explicit example where the eigenvalues are not real: take your favourite cuspidal eigenform $f$ of level $N$ and a prime $p$ not dividing $N$; then there will be two eigenforms of level $pN$ with the same eigenvalues as $f$ away from $p$, and their Hecke eigenvalues at $p$ are the roots of $X^2 - a_p(f) X + p^{k - 1}$, which are never real. 
