Towards a generalized Maeda conjecture for modular forms with quadratic nebentypus

Understanding the asymptotic behavior of the number of Galois orbits of newforms in $S_k(\Gamma_0(N), \Psi)$ as the weight increases is a central problem motivated by Maeda's conjecture. For trivial nebentypus, prior work of Dieulefait, Pacetti, and Tsaknias established a lower bound for the number of non-CM Galois orbits using local inertial types and Atkin-Lehner signs as invariants. We extend this framework to newforms with non-trivial quadratic nebentypus. On the local side, the quadratic nebentypus imposes strict central character constraints, and we explicitly determine the number of Galois orbits of admissible local inertial types. We then establish the Galois equivariance of Atkin-Li pseudo-eigenvalues, which serves as a second global invariant when taken up to a natural equivalence relation. Using existence results for newforms with prescribed local invariants, we obtain a lower bound for the number of non-CM Galois orbits for sufficiently large weights (with conditions on $N$) by counting compatible pairs of these invariants. Finally, computations in small weights reveal a strict inequality in the quadratic nebentypus setting, indicating that certain local equivalences are not realized globally by Galois conjugation over the coefficient field of the modular form.