\label{Kleinian}\end{aligned}$$

Following [@eil16] introduce co-homology basis of Klein $$\begin{aligned}
\mathrm{d}{u} (x,y), \quad  \mathrm{d}{r} (x,y)
-\frac{1}{8N_g}\Lambda_g\mathrm{d}{u} (x,y)\label{Kleinbasis}\end{aligned}$$ with constant matrix, $\Lambda_g=\Lambda_g({\lambda})$ given by (\[Lambda1\],\[Lambda2\]). Therefore we proved

$\varkappa$-matrix is represented in the modular form (\[Kleinian\]) in the co-homology basis (\[Kleinbasis\]).

Formula (\[Kleinian\]) first appears in F.Klein ([@klein886], [@klein888]), it was recently revisited in a more general context by Korotkin and Shramchenko ([@ksh12]) who extended representation for $\varkappa$ to non-hyperelliptic curves. Correspondence of this representation to the co-homology basis to the best knowledge of the authors was not earlier discussed.

Rewrite formula (\[Kleinian\]) in equivalent form, $$\begin{aligned}
\omega^T\eta=-\frac{1}{4N_g}
\sum_{N_g\;\text{even}\;[\varepsilon]}   \frac{1}{\theta[\varepsilon]}
\left(\begin{array}{ccc}
\theta_{1,1}[\varepsilon]&\cdots&\theta_{1,g}[\varepsilon]\\
\vdots&\cdots&\vdots\\
\theta_{1,g}[\varepsilon]&\cdots&\theta_{g,g}[\varepsilon]\end{array}\right)\label{omegaeta}\end{aligned}$$ where $\omega,\eta$ are half-periods of holomorphic and meromorphic differentials in Kleinian basis.

For the Weierstrass cubic $y^2=4x^3-g_2x-g_3$ (\[omegaeta\]) represents Weierstrass relation (\[SecondKind\]).

At $g=2$ (\[omegaeta\]) can be written in the form $$\begin{aligned}
\omega^T\eta = -\frac{\imath \pi}{10} \left(\begin{array}{cc}
\partial_{\tau_{1,1}} & \partial_{\tau_{1,2}}\\
\partial_{\tau_{1,2}} & \partial_{\tau_{2,2}}
 \end{array}  \right)\; \mathrm{ln}\; \chi_{5}\end{aligned}$$ where $\chi_5$ is relative invariant of weight 5, $$\begin{aligned}
\chi_{5} = \prod_{ 10\;\text{even}\; [\varepsilon] } \theta[\varepsilon]\end{aligned}$$

Worth to mention how equations of KdV flows looks in both bases. For example at $g=2$ and curve $y^2=4x^5+\lambda_4x^4+\ldots+\lambda_0$ in Baker basis we got [@bel997] $$\begin{aligned}
\begin{split}
\wp_{2222}&=6\wp_{2,2}^2+4\wp_{1,2}+\lambda_4\wp_{2,2}+\frac12\lambda_3   \\
\wp_{1222}&=6\wp_{2,2}\wp_{1,2}-2\wp_{1,1}+\lambda_4\wp_{1,2} \end{split}\end{aligned}$$ In Kleinian basis the 