  An analytic expression for the scalar quasinormal modes of the generic,
spinning Kerr-$\mathrm{AdS_5}$ black holes was previously proposed by the
authors in ref. 1, in terms of transcendental equations involving the
Painlev\'e VI (PVI) $\tau$ function. In this work, we carry out a numerical
investigation of the modes for generic rotation parameters, comparing
implementations of expansions for the PVI $\tau$ function both in terms of
conformal blocks (Nekrasov functions) and Fredholm determinants. We compare the
results with standard numerical methods for the subcase of Schwarzschild black
holes. We then derive asymptotic formulas for the angular eigenvalues and the
quasinormal modes in the small black hole limit for generic scalar mass and
discuss, both numerically and analytically, the appearance of superradiant
modes.
