  We give a proof of a formula for the trace of self-braidings (in an arbitrary
channel) in UMTCs which first appeared in the context of rational conformal
field theories (CFTs). The trace is another invariant for UMTCs which depends
only on modular data, and contains the expression of the Frobenius-Schur
indicator as a special case. Furthermore, we discuss some applications of the
trace formula to the realizability problem of modular data and to the
classification of UMTCs.
