Eichler–Shimura–Manin correspondence for newforms of levels 2 and 3

Let $k$ be an even integer. For $N=2,3$, let $\mathcal S_k^{\mathrm{new}}(\Gamma_0(N))^{\pm}$ denote the subspaces of newforms on which the Atkin–Lehner involution acts by $\pm1$. Let $W_{2,\mathrm{new},\pm}^{\mathrm{ev},0}$ and $W_{3,\mathrm{new},\pm}^{\mathrm{ev},0}$ be the polynomial spaces defined by

$$W_{2,\mathrm{new},\pm}^{\mathrm{ev},0}=\left\{p(x,y)\in\mathbb C[x,y]\ \middle|\ \begin{array}{l}-p(y,x)-p(y,x+y)+p(x,x+y)=-p(x,y),\\-p(y,2x)=\pm 2^{\frac{k-2}{2}}p(x,y)\end{array}\right\},$$

and

$$W_{3,\mathrm{new},\pm}^{\mathrm{ev},0}=\left\{p(x,y)\in\mathbb C[x,y]\ \middle|\ \begin{array}{l}-p(y,x)-p(y,x+y)+p(x,x+y)-p(y,x-y)+p(x,x-y)=-p(x,y),\\-p(y,3x)=\pm 3^{\frac{k-2}{2}}p(x,y)\end{array}\right\}.$$

Eichler–Shimura–Manin correspondence. There are isomorphisms defined over $\mathbb C$

$$\mathcal S_k^{\mathrm{new}}(\Gamma_0(2))^{\pm}\cong W_{2,\mathrm{new},\pm}^{\mathrm{ev},0},\qquad \mathcal S_k^{\mathrm{new}}(\Gamma_0(3))^{\pm}\cong W_{3,\mathrm{new},\pm}^{\mathrm{ev},0}.$$

These conjectures arise from equations observed for restricted even period polynomials of newforms at levels $2$ and $3$, extending the correspondence between modular forms and period-polynomial spaces. The source reports them as natural conjectures but gives no resolution.