  We find a family of complex saddle-points at large N of the matrix model for
the superconformal index of SU(N) N=4 super Yang-Mills theory on $S^3 \times
S^1$ with one chemical potential $\tau$. The saddle-point configurations are
labelled by points $(m,n)$ on the lattice $\Lambda_\tau= \mathbb{Z} \tau
+\mathbb{Z}$ with $\text{gcd}(m,n)=1$. The eigenvalues at a given saddle are
uniformly distributed along a string winding $(m,n)$ times along the $(A,B)$
cycles of the torus $\mathbb{C}/\Lambda_\tau$. The action of the matrix model
extended to the torus is closely related to the Bloch-Wigner elliptic
dilogarithm, and the related Bloch formula allows us to calculate the action at
the saddle-points in terms of real-analytic Eisenstein series. The actions of
$(0,1)$ and $(1,0)$ agree with that of pure AdS$_5$ and the supersymmetric
AdS$_5$ black hole, respectively. The black hole saddle dominates the canonical
ensemble when $\tau$ is close to the origin, and there are new saddles that
dominate when $\tau$ approaches rational points. The extension of the action in
terms of modular forms leads to a simple treatment of the Cardy-like limit
$\tau\to 0$.
