Let $C$ be an hyper elliptic curve.
It is known that $H^0(C,Ω_C/ \Bbb{Q})$, space of holomorphic $1$ form on $C$ has basis(sometimes called 'Hermite basis') and can be expanded by local parameter $t$.
I'm stuck with calculating concrete examples.
For example, let $C: y^2=(x^2+1)(x^4-8x^3+2x^2+8x+1)$
Let take basis {$ω_1=(4-x)dx/y, ω_2=-dx/y$} of $H^0(C,Ω_C/ \Bbb{Q})$.
By general theory, it is known that $ω_1=(t+c_{1,3}t^3＋・・・)dt/t$・・・① with respect to local parameter $t=1/x$. How can I calculate coefficients $c_{1,3}$,・・・?
Expand ① with $x=1/t$ and $y^2=(x^2+1)(x^4-8x^3+2x^2+8x+1)$, I got $ω_1=(4-1/t^2)dt/\sqrt{(1+1/t^2)(1-8t+2t^2+8t^3+t^4)} $, but this does not much ①, I may misunderstand the way.
These facts are written in $368p$ of https://www.ms.u-tokyo.ac.jp/journal/pdf/jms210205.pdf
Background : The purpose of my question is to find what $a_{23}=24- \sharp J(C)( \Bbb{F_{23}})$ is. In order to calculate Hasse Witt matrix, I need to find $c_{1,23}$ of ①.