  Character sumrules associated with the realization of the $N=4$
superconformal algebra $\At$ on manifolds corresponding to the group cosets
$SU(3)_{\ktp }/U(1)$ are derived and developed as an important tool in
obtaining the modular properties of $\At$ characters as well as information on
certain extensions of that algebra. Their structure strongly suggests the
existence of rational conformal field theories with central charges in the
range $1 \le c\le 4$. The corresponding characters appear in the massive sector
of the sumrules and are completely specified in terms of the characters for the
parafermionic theory $SU(3)/(SU(2)\times U(1))$ and in terms of the branching
functions of massless $\At$ characters into $SU(2)_{\ktp }\times SU(2)_1$
characters.
