  We use the single particle excitation energies and the completeness rules of
the 3-state anti-ferromagnetic Potts chain, which have been obtained from
Bethe's equation, to compute the modular invariant partition function. This
provides a fermionic construction for the branching functions of the $D_4$
representation of $Z_4$ parafermions which complements the previous bosonic
constructions. It is found that there are oscillations in some of the
correlations and a new connection with the field theory of the Lee-Yang edge is
presented.
