  Type-II multiferroics, in which the magnetic order breaks inversion symmetry,
are appealing for both fundamental and applied research due their intrinsic
coupling between magnetic and electrical orders. Using first-principles
calculations we study the ground state magnetic behaviour of
Ba$_7$Mn$_4$O$_{15}$ which has been classified as a type-II multiferroic in
recent experiments. Our constrained moment calculations with the proposed
experimental magnetic structure shows the spontaneous emergence of a polar mode
giving rise to an electrical polarisation comparable to other known type-II
multiferroics. When the constraints on the magnetic moments are removed, the
spins self-consistently relax into a canted antiferromagnetic ground state
configuration where two magnetic modes transforming as distinct irreducible
representations coexist. While the dominant magnetic mode matches well with the
previous experimental observations, the second mode is found to possess a
different character resulting in a non-polar ground state. Interestingly, the
non-polar magnetic ground state exhibits a significantly strong linear
magnetoelectric coupling comparable to the well-known multiferroic BiFeO$_3$,
suggesting strategies to design new linear magnetoelectrics.
