Existence of nonelliptic mod l Galois representations for every l > 5

For l = 3 and 5 it is known that every odd, irreducible, two-dimensional representation of Gal((Q) over bar /Q) with values in F, and determinant equal to the cyclotomic character must "come from" the l-torsion points of an elliptic curve defined over Q. We prove, by giving concrete counter-examples, that this result is false for every prime l > 5.