Suppose f is a newform (with coefficients generating some number field E), and $rho_{f,lambda}: {rm Gal}(overline{mathbb{Q}} / mathbb{Q}) to {rm GL}_2(E_lambda)$ the associated Galois rep (for some prime $lambda$ of E). When does $rho$ have open image in ${rm GL}_2(E_lambda)$?
This clearly isn't the case if f has weight 1, or if f is of CM type; and I gather that it's a theorem of Serre that if f is attached to an elliptic curve, then not having CM is sufficient. What's known about this question in general?
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