  We compute the superconformal index of the $\mathcal{N}=4$ $SU(N)$ Yang-Mills
theory through a residue calculation. The method is similar in spirit to the
Bethe Ansatz formalism, except that all poles are explicitly known, and we do
not require specialization of any of the chemical potentials. Our expression
for the index allows us to revisit the Cardy limit using modular properties of
four-dimensional supersymmetric partition functions. We find that all residues
contribute at leading order in the Cardy limit. In a specific region of flavour
chemical potential space, close to the two unrefined points, in fact all
residues contribute universally. These universal residues precisely agree with
the entropy functions of the asymptotically AdS$_5$ black hole and its "twin
saddle" respectively. Finally, we discuss how our formula is suited to study
the implications of four-dimensional modularity for the index beyond the Cardy
limit.
