Modularity of certain 2-dimensional mod p^n representations of Gal(Qbar/Q

For an odd rational prime p and integer n>1, we consider certain continuous representations rho_n of G_Q into GL_2(Z/p^nZ) with fixed determinant, whose local restrictions "look" like they arise from modular Galois representations, and whose mod p reductions are odd and irreducible. Under suitable hypotheses on the size of their images, we use deformation theory to lift rho_n to rho in characteristic 0. We then invoke a modularity lifting theorem of Skinner-Wiles to show that rho is modular