Complex multiplication (reviewed)

A modular form $f$ for $\GL(2)$ over a number field $F$ has complex multiplication (CM) if it is equal to its twist by a nontrivial Hecke character $\chi$ over $F$, or equivalently \[ a_{\frak{p}} = \chi(\frak{p}) a_{\frak{p}} \] for all good primes $\frak{p} \nmid \frak{N}$. Such a character $\chi$ is necessarily quadratic, and \(a_{\frak{p}} = 0\) for all good primes \(\frak{p}\) such that \(\chi(\frak{p}) = -1\).