can write

$$\begin{aligned}
\begin{split}
\varkappa_{i,j} &=\frac{1}{8N_g} \Lambda_{i,j}- \frac{1}{2 N_g}
\sum_{  \text{All even non-singular} \;\;  [\varepsilon]  }  \frac{\partial^2_{{U}_i,{U_j}}
 \theta[\varepsilon_{\mathcal{I}_0}]({0})}{\theta[\varepsilon_{\mathcal{I}_0}]({0})}\\
\end{split}\end{aligned}$$

where $$\begin{aligned}
\Lambda_{i,j}=-4\sum_{  \text{All partitions} \; \mathcal{I}_0  } \wp_{i,j}({\Omega}_{\mathcal{I}_0})\end{aligned}$$ Denote by $\Lambda_g$ symmetric matrix $$\Lambda_g = ( \Lambda_{i,j} )_{i,j=1,\ldots,g} \label{Lambda1}$$

Entries $\Lambda_{k,j}$ at $k\leq j$ to the symmetric matrix $\Lambda$ are given by the formula $$\begin{aligned}
\begin{split}
\Lambda_{k,j}&= \lambda_{k+j}  \frac{ \left( \begin{array}{c} 2g+1\\g \end{array}  \right)  }
{ \left( \begin{array}{c} 2g+1\\2g+1-k-j \end{array}  \right)  }
\sum_{n=1}^k n \left[  \left( \begin{array}{c} g\\ g-k+n \end{array}  \right) \left( \begin{array}{c} g+1\\ g-j-n+1 \end{array}  \right)  \right. \\&\hskip 5.5cm + \left.\left( \begin{array}{c} g \\g-j-n \end{array}  \right)
  \left( \begin{array}{c} g+1\\ g-k+n+1 \end{array}  \right)  \right]\end{split}\label{Lambda2}\end{aligned}$$

Execute summation in (\[formula\]) and find that each $\Lambda_{k,j}$ proportional to $\lambda_{k+j}$ with integer coefficient.

Matrix $\Lambda_g$ exhibits interesting properties of sum of anti-diagonal elements implemented at derivations in [@eil16], $$\begin{aligned}
\sum_{i,j, \;i+j=k}\Lambda_{g;i,j}=\lambda_k \frac{N_g}{4g+2}
\left[ \frac12 k(2g+2-k)+\frac14(2g+1)( (-1)^k-1 ) \right]\end{aligned}$$

Lower genera examples of matrix $\Lambda$ were given in [@ehkklp12], [@eil16], but method implemented there unable to get expressions for $\Lambda$ at big genera.

At $g=6$ we get matrix $$\begin{aligned}
\Lambda_6= \left(  \begin{array}{cccccc} 792\lambda_2&330\lambda_3&120\lambda_4&36\lambda_5&8\lambda_6&\lambda_7\\
                                       330\lambda_3&1080\lambda_4&492\lambda_5&184\lambda_6&51\lambda_7&8\lambda_8\\
                                       120\lambda_4&492\lambda_5&1200\lambda_6&542\lambda_7&184\lambda_8&36\lambda_9\\
                                       36\lambda_5&184\lambda_6&542\lambda_7&1200\lambda_8&492\lambda_9&120\lambda_{10}\\
                                       8\lambda_6&51\lambda_7&184\lambda_8&492\lambda_9&1080\lambda_{10}&330\lambda_{11}\\
                                       \lambda_7&8\lambda_8&36\lambda_9&120\lambda_{10}&330\lambda_{11}&792\lambda_{12}
\end{array}    \right)\end{aligned}$$

Collecting all these together we get the following

$\varkappa$-matrix defining multi-variate $\sigma$-function admits the following 