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By developing an algorithm for evaluating the basis states for the compositefermions with negative effective magnetic field, we perform thecomposite-fermion-diagonalization study for the fully spin-polarized fractionalquantum Hall states at the filling factors $\nu = 3/10$, 4/13, and 5/17 in therange $2/7 <\nu < 1/3$. These observed states correspond to partially filledsecond effective Landau level, for the composite fermions carrying fourvortices, with filling factor $\bar{\nu} = 1/2$, 1/3, and 2/3 respectively,analogous to the previously studied states of composite fermions with twoattached vortices in the range $1/3 <\nu <2/5$. We show that the character ofthese states in the range $2/7 <\nu < 1/3$ replicates the same for the statesin the range $1/3 <\nu <2/5$ having identical $\bar{\nu}$: Chiral p-wavepairing with anti-Pfaffian correlation of composite fermions carrying sixquantized vortices produces incompressible state at $\nu = 3/10$; anunconventional interaction between composite fermions, resulting from thesuppression of fermion pairs with relative angular momentum three and producingfractional quantum Hall effect of composite fermions in the second effectiveLandau level with $\bar{\nu} =1/3$ and its particle-hole conjugate fillingfactor 2/3, reproduces incompressible states at $4/13$ and $5/17$ fillingfactors. We further estimate the thermodynamic limit of the ground stateenergies and calculate the lowest energy gap for neutral collective excitationsof these states.

Incompressible States of the Interacting Composite Fermions in Negative Effective Magnetic Fields at $\nu=4/13$, 5/17, and 3/10