Observed slope and eigenvalue relations for $N=7$

For $p=2$ and $N=7$, let $\alpha_{\mathcal{L}}^\pm(k,2,7)$ denote the multisets of slopes on the $\pm1$ Atkin–Lehner eigenspaces, let $\alpha_{\mathcal{L}}(k,2,7)$ be their combined multiset, let $\alpha_{\mathcal{L}}(k,2,3)$ be the combined slope multiset for $N=3$, and let $\varepsilon_W(k,2,7)$ denote the corresponding Atkin–Lehner eigenvalues. Observed slope and eigenvalue relations for $N=7$. The source records the following observations: (i) $\alpha_{\mathcal{L}}^+(k,2,7)=\alpha_{\mathcal{L}}^-(k,2,7)$ for all $k\geq4$; (ii) the slopes appearing in $\alpha_{\mathcal{L}}(k,2,3)$ also appear in $\alpha_{\mathcal{L}}^-(k,2,7)$; and (iii), for $k\in4\mathbb{Z}$ with $k\geq8$,

$$\alpha_{\mathcal{L}}(k,2,7)=\alpha_{\mathcal{L}}(k+2,2,7),\qquad \varepsilon_W(k,2,7)=-\varepsilon_W(k+2,2,7).$$

These claims are stated as computational observations based on the displayed data, and the source gives no proof or resolution.