# On the number of Galois orbits of newforms

### Luis Victor Dieulefait

Universitat de Barcelona, Spain### Ariel Pacetti

Universidad Nacional de Córdoba, Argentina### Panagiotis Tsaknias

Purley, UK

## Abstract

Counting the number of Galois orbits of newforms in $S_{k}(Γ_{0}(N))$ 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 $k≥16$. 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 $Γ_{0}(N)$ 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.

## Cite this article

Luis Victor Dieulefait, Ariel Pacetti, Panagiotis Tsaknias, On the number of Galois orbits of newforms. J. Eur. Math. Soc. 23 (2021), no. 8, pp. 2833–2860

DOI 10.4171/JEMS/1073