  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.
