  It is shown that every weak Jacobi form of weight zero and index one on a
congruence subgroup of the full Jacobi group can be decomposed into $N=4$
superconformal characters. Additionally, a simple expression for the mock
modular form determining the superconformal character coefficients is obtained,
as well as a universal completion structure. Along the way, a useful
vector-valued mock modular form is also found and studied. These results are
applied to analyze some Jacobi trace functions associated to super vertex
operator algebras and a distinguished sector.
