Connections between Double Zeta Values relative to $\mu_N$, Hecke Operators $T_N$, and Newforms of Level $\Gamma_0(N)$ for $N=2,3$

TL;DR

This paper explores the relationships between double zeta values, Hecke operators, and newforms of specific levels, extending known results and proposing a conjecture on the Eichler-Shimura-Manin correspondence.

Contribution

It generalizes previous work by Baumard and Schneps, establishing new connections and conjectures in the theory of modular forms and multiple zeta values.

Findings

Generalized connections between double zeta values and Hecke operators for levels 2 and 3.

Proposed a conjecture relating newforms and period polynomial spaces.

Extended the understanding of the Eichler-Shimura-Manin correspondence in this context.

Abstract

In this paper, we will study various connections between double zeta values relative to $\mu_N$, Hecke operators $T_N$, and newforms of level $\Gamma_0(N)$ for $N=2,3$. Those various connections generalize the well-known of Baumard and Schneps. We also give a conjecture about the Eichler-Shimura-Manin correspondence between $\mathcal{S}_k^{\textrm{new}}(\Gamma_0(2))^{\pm}$, $\mathcal{S}_k^{\textrm{new}}(\Gamma_0(3))^{\pm}$ and some period polynomial spaces.