  Exact diagonalization of a two-dimensional electron gas in a strong magnetic
field in the disk geometry shows that there exists a filling factor range in
the second Landau level where the states significantly differ from those in the
lowest Landau level. We show that the difference arises because the interaction
between composite fermions is not negligible in higher Landau levels, as
indicated by a substantial mixing between composite-fermion Landau-like levels,
or lambda levels. We find that the exact ground state is well reproduced by
composite fermion theory with lambda level mixing incorporated at the lowest
level of approximation. Using the same variational approach in the spherical
geometry we estimate the excitation gap at filling 1/3 in the second Landau
level (which corresponds to 7/3 of experiment).
