  Here we follow on the proposed generalization of Maeda's conjecture made in
[2]. We report on computations that suggest a relation between the number of
local types and the number of non-CM newform Galois orbits. We extend the
conjecture into spaces with non-trivial Nebentypus and provide a formula for
the number of non-CM orbits for all levels and trivial Nebentypus. We also
provide some numerical evidence towards further generalizations of this
conjecture to totally real fields as well as further strengthening of it by
proposing a structure for the corresponding Galois groups.
