A Possible Generalization of Maeda’s Conjecture

Abstract

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.

On the gaps between non-zero Fourier coefficients of cusp forms of higher weight

“…Now we shall give an example of a Hecke eigenform f of weight k > 2 and level N > 1 with CM and satisfy the condition that i f (n) ≪ n Proof. From the tables in [Tsa14], we see that the cuspidal Hecke eigenform f E in (4.6) has CM. By Theorem 3.3, the cuspidal eigenform f E satisfies (4.8).…”

“…Using SAGE, we have checked that these indeed hold and hence all these 3 cuspidal eigenforms in S new 4 (32) satisfies the identity (4.8), by Theorem 4.2. From the tables in [Tsa14], we see that, out of these 3 eigenforms f Hence, there is an example of a Hecke eigenform f of weight k > 2 and level N > 1 with CM and satisfy the condition that i f (n) ≪ n 1 4 .…”

A generalization of Ramanujan's congruence to modular forms of prime level

“…Assuming a conjecture of Maeda, the Hecke eigenforms of full level form a single Galois orbit, so all of them satisfy (1.1) modulo conjugate ideals. Similarly, a generalization of Maeda's conjecture due to Tsaknias [33] states that the newforms in S k (p) form two Galois orbits for sufficiently large k, the forms in each orbit sharing the same Atkin-Lehner eigenvalue. This would imply that all newforms in S k (p) satisfy congruence (1.5), when primes ℓ as in Theorem 1 exist.…”

Refined dimensions of cusp forms, and equidistribution and bias of signs

“…This gives a lower bound on the number of Galois orbits in S new k (N ). Conjecturally, this lower bound equals the number of Galois orbits for sufficiently large k (see [Tsa14]). Thus our bias of root numbers suggests that Galois orbits tend to be slightly larger for newforms with root number +1.…”