  By developing an algorithm for evaluating the basis states for the composite
fermions with negative effective magnetic field, we perform the
composite-fermion-diagonalization study for the fully spin-polarized fractional
quantum Hall states at the filling factors $\nu = 3/10$, 4/13, and 5/17 in the
range $2/7 <\nu < 1/3$. These observed states correspond to partially filled
second effective Landau level, for the composite fermions carrying four
vortices, with filling factor $\bar{\nu} = 1/2$, 1/3, and 2/3 respectively,
analogous to the previously studied states of composite fermions with two
attached vortices in the range $1/3 <\nu <2/5$. We show that the character of
these states in the range $2/7 <\nu < 1/3$ replicates the same for the states
in the range $1/3 <\nu <2/5$ having identical $\bar{\nu}$: Chiral p-wave
pairing with anti-Pfaffian correlation of composite fermions carrying six
quantized vortices produces incompressible state at $\nu = 3/10$; an
unconventional interaction between composite fermions, resulting from the
suppression of fermion pairs with relative angular momentum three and producing
fractional quantum Hall effect of composite fermions in the second effective
Landau level with $\bar{\nu} =1/3$ and its particle-hole conjugate filling
factor 2/3, reproduces incompressible states at $4/13$ and $5/17$ filling
factors. We further estimate the thermodynamic limit of the ground state
energies and calculate the lowest energy gap for neutral collective excitations
of these states.
