  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.
