  Conformal field theory gives quite detailed predictions for the low energy
spectrum and scaling exponents of a massless Luttinger liquid at generic
filling in the presence of an impurity. While these predictions were verified
for half-filled systems, there was till now no analysis away from this
particular filling. Here, we fill in this gap by numerically investigating a
quarter-filled system using the density matrix renormalization group technique.
Our results confirm conformal field theory predictions, and suggest that they
are indeed valid for arbitrary fillings.
