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We study the superconfomal index of four-dimensional toric quiver gaugetheories using a Bethe Ansatz approach recently applied by Benini and Milan.Relying on a particular set of solutions to the corresponding Bethe Ansatzequations we evaluate the superconformal index in the large $N$ limit, thusavoiding to take any Cardy-like limit. We present explicit results for theoriesarising as a stack of $N$ D3 branes at the tip of toric Calabi-Yau cones: theconifold theory, the suspended pinch point gauge theory, the first del Pezzotheory and $Y^{p,q}$ quiver gauge theories. For a suitable choice of thechemical potentials of the theory we find agreement with predictions made forthe same theories in the Cardy-like limit. However, for other regions of thedomain of chemical potentials the superconformal index is modified andconsequently the associated black hole entropy receives corrections. We workout explicitly the simple case of the conifold theory.

Microstate Counting via Bethe Ans\"{a}tze in the 4d ${\cal N}=1$ Superconformal Index