call a *seeded relation.* This is one of the key ideas in finding the formula in [@DG 5.25] concerning $d(w,N)$.

First, we know there are $N-1$ MPVs of weight 1 and level $N$: $$L_N(1|j)=-\log(1-\mu^j), \qquad 0<j<N ,$$ where $\mu=\mu_N=\exp(2\pi\sqrt{-1}/N)$ as before. Here we have taken $\C-(-\infty,0]$ as the principle domain of the logarithm. Further, it follows from the motivic theory of classical polylogs developed by Deligne and Beilinson and the Borel’s theorem (see [@G1 Thm. 2.1]) that the $\Q$-dimension of $\MPV(1,N)$ is $$d(1,N)=\dim K_1(\Z[\mu_N][1/N])\otimes \Q+1=\varphi(N)/2+\nu(N),$$ where $\varphi$ is the Euler’s totient function and $\nu(N)$ is the number of distinct prime factors of $N$. Hence there are many linear relations among $L_N(1|j)$. For instance, if $j<N/2$ then we have the symmetric relation $$-\log(1-\mu^j)=-\log(1-\mu^{N-j})-\log(-\mu^j)=
 -\log(1-\mu^{N-j})+\frac{N-2j}{N}\pi\sqrt{-1}.$$ Thus for all $1<j<N/2$ $$\label{equ:symrel}
(N-2)(L_N(1|j)-L_N(1|N-j))=(N-2j)(L_N(1|1)-L_N(1|N-1)).$$ Further, from [@Bass (B)] for any divisor $d$ of $N$ and $1\le a<d':=N/d$ we have the distribution relation $$\label{equ:prodrel}
    \sum_{0\le j<d} L_N(1|a+jd')=  L_N(1|ad).$$ It follows from the main result of Bass [@Bass] corrected by Ennola [@En] that all the linear relations between $L_N(1|j)$ are consequences of and . Hence the seeded relations have the following forms in words: for all $w\in \fA^0$ $$\label{equ:seed}
\left\{\aligned
(N-2)Z(y_{1,j}*w-y_{1,-j}*w)=&(N-2j)(Z(y_{1,1}*w-y_{1,-1}*w),\\
\sum_{0\le j<d} Z(y_{1,a+jd'}*w)=&  Z(y_{1,ad}*w).
\endaligned\right.$$

Regularized 