Let $f$ be a modular form (say of weight 2) and let $S$ be the set of primes $p$, the reductions modulo $p$ of $\rho_{f,p}$ restricted to $G_p$ splits. Equivalently, $S$ is the set of prime $p$ such that $f$ has a companion form modulo $p$. Now if $f$ has CM then $S$ is an infinite set. Is the converse true? That is, does there exist a modular form $f$ without CM such that $S$ is infinite?
I think that there are results that says that if the $p$-adic representation $\rho_{f,p}$ restricted to $G_p$ is split then $f$ has CM, but I don't know how (or if) that can be helpful for this problem.
