In this paper we make a proposal for the solution to a long-standing problem — the asymptotic expansions of the modular S-transform of a generalised Gibbs ensemble (GGE) in a theory with \( {\mathcal{W}}_3 \) symmetry where the GGE includes the first non-trivial charge. Equivalently, we give a proposal for the modular S-transform of traces of arbitrary powers of the zero mode W0. We provide evidence in the form of exact results using Zhu’s recursion, results obtained using conjectured results for Verma modules, and exact results for the particular value c = –2. We expect these have generalisations to other symmetry algebras/hierarchies such as the Virasoro algebra/KdV charges, and to GGEs with arbitrary finite sets of charges.
