Structure and bases of modular space sequences $(M_{2k}(\Gamma_0(N)))_{k\in \mathbb{N}^*}$ and $(S_{2k}(\Gamma_0(N)))_{k\in \mathbb{N}^*}$. Part II: a modular butterfly hunt

TL;DR

This paper explores the structure and explicit bases of modular form spaces for levels 1 to 10, introducing the concept of strong modular units to facilitate a detailed analysis and basis construction.

Contribution

It introduces the notion of level N strong modular units and provides explicit bases for modular form spaces for levels 1 to 10.

Findings

Explicit bases for modular form spaces at levels 1 to 10.

Identification of strong modular units for these levels.

Structural insights into modular forms related to these levels.

Abstract

In the first part of this article, which contains three of them, we have identified the notion of level $N$ strong modular unit. It enabled us to structure the modular forms family $(M_{2k}(\Gamma_0(N)))_{k\in \mathbb{N}^*}$ and to propose the explicit bases for these spaces. It is in this perspective that we wrote this second part where the structure and explicit bases are proposed when $1\leq N \leq 10$.