An explicit representation of primitive forms

TL;DR

This paper explicitly describes primitive forms for levels 1, 2, 3, 4, 6, 8, 9, focusing on Galois conjugacy classes with small cardinality using Eisenstein series, with some computations pending for level 6.

Contribution

It provides explicit formulas for primitive forms in terms of Eisenstein series for multiple levels, advancing the understanding of their Galois conjugacy classes.

Findings

Explicit descriptions for levels 1, 2, 3, 4, 8, 9

Partial results for level 6

Decomposition into Galois conjugacy classes with small cardinality

Abstract

The sets of primitive foms may be decomposed into some Galois conjugacy classes. The purpose of this paper is to write down all of such classes with cardinal 1 or 2, explicitly in terms of some Eisenstein series, for level 1,2,3,4,6,8,9. For level 6, some calculations have not yet been completed.