  Fractional quantum Hall states at a half-filled Landau level are believed to
carry an integer number $\mathcal{C}$ of chiral Majorana edge modes, reflected
in their thermal Hall conductivity. We show that this number determines the
primary series of Abelian fractional quantum Hall states that emerge above and
below the half-filling point. On a particular side of half-filling each series
may originate from two consecutive values of $\mathcal{C}$, but the combination
of the series above and below half-filling uniquely identifies $\mathcal{C}$.
We analyze these states both by a hierarchy approach and by a composite fermion
approach. In the latter, we map electrons near a half-filled Landau level to
composite fermions at a weak magnetic field and show that a bosonic integer
quantum Hall state is formed by pairs of composite fermions and plays a crucial
role in the state's Hall conductivity.
