  There has been a recent progress in understanding the chiral ring of 3d
$\mathcal{N}=4$ superconformal gauge theories by explicitly constructing an
exact generating function (Hilbert series) counting BPS operators on the
Coulomb branch. In this paper we introduce Coulomb branch Hilbert series in the
presence of background magnetic charges for flavor symmetries, which are useful
for computing the Hilbert series of more general theories through gluing
techniques. We find a simple formula of the Hilbert series with background
magnetic charges for $T_\rho(G)$ theories in terms of Hall-Littlewood
polynomials. Here $G$ is a classical group and $\rho$ is a certain partition
related to the dual group of $G$. The Hilbert series for vanishing background
magnetic charges show that Coulomb branches of $T_\rho(G)$ theories are
complete intersections. We also demonstrate that mirror symmetry maps
background magnetic charges to baryonic charges.
