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In a previous work, exact formulae and differential equations were found fortraces of powers of the zero mode in the W3 algebra. In this paper weinvestigate their modular properties, in particular we find the exact resultfor the modular transformations of traces of $W_0^n$ for n = 1, 2, 3, solvingexactly the problem studied approximately by Gaberdiel, Hartman and Jin. Wealso find modular differential equations satisfied by traces with a single$W_0$ inserted, and relate them to differential equations studied by Mathur etal. We find that, remarkably, these all seem to be related to weight 0 modularforms with expansions with non-negative integer coefficients.

Modular properties of characters of the W3 algebra