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? 



When do the Galois reps of modular forms have open image?
The answer of TSG is not correct in all cases. For a complete and detailed answer see this later [question](https://mathoverflow.net/questions/220845/when-is-the-image-of-a-2-dim-l-adic-representation-associated-to-a-modular-form?answertab=votes#tab-top) and the answer of David Loeffler (the very OP of the current question).


