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In this paper, we will study various connections between double zeta valuesrelative 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 andSchneps. We also give a conjecture about the Eichler-Shimura-Manincorrespondence between $\mathcal{S}_k^{\textrm{new}}(\Gamma_0(2))^{\pm}$,$\mathcal{S}_k^{\textrm{new}}(\Gamma_0(3))^{\pm}$ and some period polynomialspaces.

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