Generating-series conjecture for newform dimensions at levels 2 and 3

For $N=2,3$, let $\mathcal S_k^{\mathrm{new}}(\Gamma_0(N))^{\pm}$ denote the subspaces of newforms of weight $k$ and level $\Gamma_0(N)$ on which the Atkin–Lehner involution acts by $\pm1$. Generating-series conjecture. The generating series of their dimensions, for $k\geq6$, are

$$\sum_{k=6}^{\infty}\dim\bigl(\mathcal S_k^{\mathrm{new}}(\Gamma_0(2))\bigr)^+x^k=\frac{x^8}{(1-x^6)(1-x^8)},$$ $$\sum_{k=6}^{\infty}\dim\bigl(\mathcal S_k^{\mathrm{new}}(\Gamma_0(2))\bigr)^-x^k=\frac{x^2(1+x^{18})}{(1-x^8)(1-x^{12})},$$ $$\sum_{k=6}^{\infty}\dim\bigl(\mathcal S_k^{\mathrm{new}}(\Gamma_0(3))\bigr)^+x^k=\frac{x^8}{(1-x^2)(1-x^{12})},$$ $$\sum_{k=6}^{\infty}\dim\bigl(\mathcal S_k^{\mathrm{new}}(\Gamma_0(3))\bigr)^-x^k=\frac{x^6(1+x^8+x^{10}-x^{12})}{(1-x^4)(1-x^{12})}.$$

The conjecture is motivated by numerical agreement between eigenspace dimensions of the relevant matrices and dimensions of the corresponding newform spaces; the lower limit $k=6$ reflects the vanishing of the lower-weight newform spaces and the range in which the method applies. No resolution is supplied in the source.