Abstract. We report on observations we made on computational data that suggest a generalization of Maeda's conjecture regarding the number of Galois orbits of newforms of level N = 1, to higher levels. They also suggest a possible formula for this number in many of these cases.

Bianchi modular forms are automorphic forms over an imaginary quadratic field, associated to a Bianchi group. Even though modern studies of Bianchi modular forms go back to the mid 1960's, most of the fundamental problems surrounding their theory are still wide open. Only for certain types of Bianchi modular forms, which we will call non-genuine, it is possible at present to develop dimension formulas: They are (twists of) those forms which arise from elliptic cuspidal modular forms via the Langlands Base-Change procedure, or arise from a quadratic extension of the imaginary quadratic field via automorphic induction (so-called CMforms). The remaining Bianchi modular forms are what we call genuine, and they are of interest for an extension of the modularity theorem (formerly the Taniyama-Shimura conjecture, crucial in the proof of Femat's Last Theorem) to imaginary quadratic fields. In a preceding paper by Rahm and Şengün, an extreme paucity of genuine cuspidal Bianchi modular forms has been reported, but those and other computations were restricted to level One. In this paper, we are extending the formulas for the non-genuine Bianchi modular forms to deeper levels, and we are able to spot the first, rare instances of genuine forms at deeper level and heavier weight.

This article surveys modularity, level raising and level lowering questions for two-dimensional representations modulo prime powers of the absolute Galois group of the rational numbers. It contributes some new results and describes algorithms and a database of modular forms orbits and higher congruences. MS Classification (2010): 11F33, 11F80

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.

For a rational prime p ≥ 3 we show that a p-ordinary modular eigenform f of weight k ≥ 2, with p-adic Galois representation ρ f , mod p m reductions ρ f,m , and with complex multiplication (CM), is characterized by the existence of p-ordinary CM companion forms hm modulo p m for all integers m ≥ 1 in the sense that ρ f,m ∼ ρ hm ,m ⊗χ k−1 , where χ is the p-adic cyclotomic character.