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