  By applying the Atiyah-Bott-Berline-Vergne equivariant integration formula
upon double dimensional integrals, we find a way to compute the matrix integral
representations of $4d$ $\mathcal{N}=1$ superconformal indices. The final
formula allows us to easily extract analytic results in the large-rank
expansion of certain theories. As an example, we compute the leading one-loop
corrections to the effective action of the known complex saddles in those
theories. For a superconformal index of $SU(N)$ $\mathcal{N}=4$ SYM, we use the
equivariant integration formula and the Picard-Lefschetz method to show that at
large enough values of $N$, only two, among the known complex saddles, dominate
the counting of large operators i.e. of operators with charges of order $N^2$.
Contributions from other known complex saddles are present, but we show they
are exponentially suppressed in that range of charges; the smaller the charges
the less suppressed they are, and eventually, to count small operators i.e.
operators with charges smaller than $N^{2/3}$, like multi-gravitons, they can
not be neglected.
