Aha - thanks! So you mean that if $f=\sum_{n=1}^\infty c(n)q^n$, I don't take the form [ g = \sum_{n=1}^\infty \chi(n)c(n)q^n ] but I write $g$ as $g=f_\chi + g'$, where $f_\chi$ is a newform? This would make perfect sense, of course. (But it is a bit confusing that people write in several places things like $f$ has CM if and only if it is equal to a twist of itself.)