Arithmetic Constraints on Charges of Symmetric Monopoles

For Euclidean BPS monopoles with gauge group $SU(N)$, maximal symmetry breaking, and sufficiently generic spectral data, we produce necessary congruence relations that their magnetic charges must satisfy when $N\ge 3$. As such, a monopole as above with given charges can have certain subgroups of $SO(3)$ ruled out as possible groups of symmetries. In particular, the axially symmetric case may only admit non-trivial magnetic charges in the non-generic setting.