Experimental evidence for Maeda's conjecture on modular forms

Abstract

We describe a computational approach to the verification of Maeda's conjecture for the Hecke operator T 2 on the space of cusp forms of level one. We provide experimental evidence for all weights less than 12 000, as well as some applications of these results. The algorithm was implemented using the mathematical software Sage, and the code and resulting data were made freely available. * We thank David Harvey for asking a question that lead us to drastically improve our Sage implementation, and David Farmer, Gabor Wiese and the referees for very useful comments.

Newforms with rational coefficients

“…In other words, it says that the Galois group of f k,1,p (x) is always S d k . The reference [GM12] also proves the strengthened conjecture for p = 2 and k ≤ 14000, and surveys other results related to the conjecture.…”

Distinguishing newforms

“…We refer the reader to [7] for a survey of results on Maeda's conjecture and a report on its verification for the operator T 2 and weights k ≤ 14 000.…”

An application of Maeda’s conjecture to the inverse Galois problem

“…Maeda's conjecture was formulated as Conjecture 1.2 in [HM97]. It has been checked up to weight 12000 (see [GM12]). We also mention that a generalisation of a weaker form of Maeda's conjecture to squarefree levels has recently been proposed by Tsaknias [Tsa12].…”