Non-Abelian quantized Hall states of electrons at filling factors 12/5 and 13/5 in the first excited Landau level

We present results of extensive numerical calculations on the ground state of electrons in the first excited (n=1) Landau level with Coulomb interactions, and including nonzero thickness effects, for filling factors 12/5 and 13/5 in the torus geometry. In a region that includes these experimentally relevant values, we find that the energy spectrum and the overlaps with the trial states support the previous hypothesis that the system is in the non-Abelian k=3 liquid phase we introduced in a previous paper.