  We study two dimensional systems with extended conformal symmetry generated
by the ${\mathcal W}_3$ algebra. These are expected to have an infinite number
of commuting conserved charges, which we refer to as the quantum Boussinesq
charges. We compute the eigenvalues of the quantum Boussinesq charges in both
the vacuum and first excited states of the higher spin module through the
ODE/IM correspondence. By studying the higher spin conformal field theory on
the torus, we also calculate thermal correlators involving the energy-momentum
tensor and the spin-3 current by making use of the Zhu recursion relations. By
combining these results, we show that it is possible to derive the current
densities, whose integrals are the quantum Boussinesq charges. We also evaluate
the thermal expectation values of the conserved charges, and show that these
are quasi-modular differential operators acting on the character of the higher
spin module.
