  We compute the Hilbert series of the graded algebra of real regular functions
on the symplectic quotient associated to an $\operatorname{SU}_2$-module and
give an explicit expression for the first nonzero coefficient of the Laurent
expansion of the Hilbert series at $t = 1$. Our expression for the Hilbert
series indicates an algorithm to compute it, and we give the output of this
algorithm for representations of dimension at most $10$. Along the way, we
compute the Hilbert series of the module of covariants of an arbitrary
$\operatorname{Sl}_2$- or $\operatorname{SU}_2$-module as well its first three
Laurent coefficients.
