It has been recently understood that BPS AdS5 black holes are encoded in a certain partition function (known as the 4d superconformal index) of the N=4 super-Yang-Mills, in the limit where the "complexified temperature" approaches 2\pi i. In this seminar we discuss limits where the complexified temperature approaches rational multiples of 2\pi i (hence the associated fugacity nears a root of unity). We will find infinitely-many saddles different from the black holes, and discuss the analogy with the better-known AdS3/CFT2 setting where also infinitely-many saddles arise in a so-called Farey Tail expansion of the 2d superconformal index.