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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?

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    $\begingroup$ Can you not take $F$ to be supercuspidal at some prime dividing $n$? Or twist by a ramified character? (I don't know about Hida families.) $\endgroup$
    – Kimball
    Commented Nov 1, 2023 at 23:57
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    $\begingroup$ Kimball's construction can certainly be used to force $a_{n} = 0$ for all $n$ divisible by $\ell$, where $\ell$ is some prime dividing the level. But if we restrict to vanishing of $a_n$'s for $n$ coprime to the level of the Hida family then I don't know, and I think the question is a very interesting one. $\endgroup$
    – David Loeffler
    Commented Nov 2, 2023 at 6:26

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