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The degeneracies of $1/4$ BPS states with unit torsion in heterotic stringtheory compactified on a six-torus are given in terms of the Fouriercoefficients of the reciprocal of the Igusa cusp Siegel modular form$\Phi_{10}$ of weight $10$. We use the symplectic symmetries of the latter toconstruct a fine-grained Rademacher type expansion which expresses these BPSdegeneracies as a regularized sum over residues of the poles of $1/\Phi_{10}$.The construction uses two distinct ${\rm SL}(2, \mathbb{Z})$ subgroups of ${\rmSp}(2, \mathbb{Z})$ which encode multiplier systems, Kloosterman sums andEichler integrals appearing therein. Additionally, it shows how the polar dataare explicitly built from the Fourier coefficients of $1/\eta^{24}$ by means ofa continued fraction structure.

Rademacher expansion of a Siegel modular form for ${\cal N}= 4$ counting