  We study novel invariants of modular categories that are beyond the modular
data, with an eye towards a simple set of complete invariants for modular
categories. Our focus is on the $W$-matrix--the quantum invariant of a colored
framed Whitehead link from the associated TQFT of a modular category. We prove
that the $W$-matrix and the set of punctured $S$-matrices are strictly beyond
the modular data $(S,T)$. Whether or not the triple $(S,T,W)$ constitutes a
complete invariant of modular categories remains an open question.
