  We study the superconformal index of 3d $\mathcal{N}=2$ superconformal field
theories on $S^1\times_{\omega} S^2$ in the Cardy-like limit where the radius
of the $S^1$ is much smaller than that of the $S^2$. We show that the first two
leading terms in this Cardy-like expansion are dictated by the Bethe Ansatz
formulation of the topologically twisted index of the same theory. We apply
this relation to 3d $\mathcal{N}=2$ holographic superconformal field theories
describing the low-energy dynamics of $N$ M2-branes and derive closed form
expressions, valid to all orders in the $1/N$ expansion, for the two leading
terms in the Cardy-like expansion of the superconformal index. We also discuss
the implications of our results for the entropy of supersymmetric Kerr-Newman
black holes in AdS$_4$ and the four-derivative corrections to 4d gauged
supergravity.
