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The entropy of $1/16$-th BPS AdS$_5$ black holes can be microscopicallyaccounted for by the superconformal index of the $\mathcal{N}=4$super-Yang-Mills theory. One way to compute this is through a Cardy-like limitof a formula for the index obtained in [1] using the "$S$-transformation" ofthe elliptic $\Gamma$ function. In this paper, we derive more general$SL(3,\mathbb{Z})$ modular properties of the elliptic $\Gamma$ function. Wethen use these properties to obtain a three integer parameter family ofgeneralized Cardy-like limits of the $\mathcal{N}=4$ superconformal index. Fromthese limits, we obtain entropy formulae that have a similar form as that ofthe original AdS$_5$ black hole, up to an overall rescaling of the entropy. Weinterpret this both on the field theory and the gravitational side. Finally, wecomment on how our work suggests a generalization of the Farey tail to fourdimensions.

$SL(3,\mathbb{Z})$ Modularity and New Cardy Limits of the $\mathcal{N}=4$ Superconformal Index