Single centred 1/4 BPS degeneracies in four-dimensional $N=4$ heterotic string theory are generated by $1/\Phi_{10}$. In this talk, we will demonstrate how its $Sp(2,\mathbb{Z})$ symmetries enable us to develop not only a Rademacher expansion for these degeneracies, but also derive the explicit structure underlying the polar states that feed into the expansion. Our results have important implications for using microscopic data to exactly compute gravity path integrals.
