TITLE: When do the Galois reps of modular forms have open image?
QUESTION [8 upvotes]: 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?

REPLY [4 votes]: The answer of TSG is not correct in all cases. For a complete and detailed answer see this later question and the answer of David Loeffler (the very OP of the current question).