  It has long been puzzling that fractional quantum Hall states in the first
excited Landau level (1LL) often differ significantly from their counterparts
in the lowest Landau level. We show that the dispersion of composite fermions
(CFs) is a deterministic factor driving the distinction. We find that CFs with
two quantized vortices in the 1LL have a non-quasiconvex dispersion.
Consequently, in the filling fraction $7/3$, CFs occupy the second
$\Lambda$-level instead of the first. The corresponding ground state wave
function, based on the CF wave function ansatz, is identified to be the
fermionic Haffnian wave function rather than the Laughlin wave function. The
conclusion is supported by numerical evidence from exact diagonalizations in
both disk and spherical geometries. Furthermore, we show that the dispersion
becomes quasiconvex in wide quantum wells or for CFs with four quantized
vortices in the filling fraction $11/5$, coinciding with observations that the
distinction between the Landau levels disappears under these circumstances.
