 split $\dim_{\mathbb Q}\ker C_{N,0}$ into $$\begin{aligned} \dim_{\mathbb Q}\ker C_{N,0}=\dim_{\mathbb Q}\ker{\prescript{t\!}{}{\left( E_{N,0}^{(0)}E_{N,0}^{(0)}E_{N,0}^{(0)}\right)}}+\dim_{\mathbb Q}\left({\operatorname{Im}}{\prescript{t\!}{}{