  The degeneracies of $1/4$ BPS states with unit torsion in heterotic string
theory compactified on a six-torus are given in terms of the Fourier
coefficients of the reciprocal of the Igusa cusp Siegel modular form
$\Phi_{10}$ of weight $10$. We use the symplectic symmetries of the latter to
construct a fine-grained Rademacher type expansion which expresses these BPS
degeneracies 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 ${\rm
Sp}(2, \mathbb{Z})$ which encode multiplier systems, Kloosterman sums and
Eichler integrals appearing therein. Additionally, it shows how the polar data
are explicitly built from the Fourier coefficients of $1/\eta^{24}$ by means of
a continued fraction structure.
