\dim_{\mathbb Q}\mathbf W_{N,0}-\dim_{\mathbb Q}\ker{\varphi^{(0)}_{0}\big|_{\mathbf W_{N,0}}}$ we obtain $$\begin{aligned} \label{eq:case0ineq0}     \sum_{N>0}\dim_{\mathbb Q}{\operatorname{Im}}{\varphi^{(0)}_{0}\big|_{\mathbf W_{N,0}}}