Open conditions for infinite multiplicity eigenvalues on elliptic curves

Let E be an elliptic curve defined over a number field K, V the complexification of the group of rational points of E over an algebraic closure L of K, and G the Galois group Gal(L/K). We show that for each root of unity w, the set of elements g in G such that w is an eigenvalue of infinite multiplicity for g acting on V has non-empty interior.