  Counting the number of Galois orbits of newforms in $S_k(\Gamma_0(N))$ and
giving some arithmetic sense to this number is an interesting open problem. The
case $N=1$ corresponds to Maeda's conjecture (still an open problem) and the
expected number of orbits in this case is 1, for any $k \ge 16$. In this
article we give local invariants of Galois orbits of newforms for general $N$
and count their number. Using an existence result of newforms with prescribed
local invariants we prove a lower bound for the number of non-CM Galois orbits
of newforms for $\Gamma_0(N)$ for large enough weight $k$ (under some technical
assumptions on $N$). Numerical evidence suggests that in most cases this lower
bound is indeed an equality, thus we leave as a Question the possibility that a
generalization of Maeda's conjecture could follow from our work. We finish the
paper with some natural generalizations of the problem and show some of the
implications that a generalization of Maeda's conjecture has.
