Connections between Double Zeta Values relative to ${\mu }_N$, Hecke Operators $T_N$, and Newforms of Level ${\Gamma }_0(N)$ for $N=2,3$

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 ${\mathcal{S}}_k^{\text{new}}({\Gamma }_0(2){)}^{\pm }$, ${\mathcal{S}}_k^{\text{new}}({\Gamma }_0(3){)}^{\pm }$ and some period polynomial spaces.