Suppose you have a Hida family with $q$-expansion $F = \sum_{n=1}^{\infty} a_n(T) q^n$, where the coefficients $a_n(T)$ are power series in $\mathbb{Z}_p [[T]]$. Assume that $F$ is a cuspidal eigenform. If there exists an integer $n$ such that $a_n(T) = 0$, does $F$ have CM?

The converse is definitely true: if $F$ has CM by an imaginary quadratic field $K$, then $a_q(T)=0$ for all primes $q$ which are nonsplit in $K$. But I cannot think of any non-CM examples of Hida families which have a vanishing Fourier coefficient. Are there any such examples?