BPS states in the Minahan-Nemeschansky $E_7$ theory

We use the method of spectral networks to calculate BPS degeneracies in the Minahan-Nemeschansky $E_7$ theory, as representations of the $E_7$ flavor symmetry. Our results provide another example of a pattern noticed earlier in the Minahan-Nemeschansky $E_6$ theory: when the electromagnetic charge is $n$ times a primitive charge, the BPS index is a positive integer multiple of $(-1{)}^{n+1}n$. We also calculate BPS degeneracies in the Minahan-Nemeschansky $E_6$ theory for larger charges than were previously computed.