  Traces of powers of the zero mode in the W3 Algebra have recently been found
to be of interest, for example in relation to Black Hole thermodynamics, and
arise as the terms in an expansion of the full characters of the algebra. We
calculate the first few such powers in two cases. Firstly, we find the traces
in the 3-state Potts model by using null vectors to derive modular differential
equations for the traces. Secondly, we calculate the exact results for Verma
module representations. We compare our two methods with each other and the
result of brute-force diagonalisation for low levels and find complete
agreement.
