Consider an elliptic curve $E\!: y^2 = x(x-1)(x-\lambda)$, where $\lambda \in k \setminus \{0, 1\}$ for some field $k$ of $\mathrm{char}(k) \neq 2$. Also consider the ample divisor $D = 2P_0$, where $P_0 = (0,0)$.
I would like to understand what are algebraic theta-functions (see, for example, "Mumford, On the Equations Defining Abelian Varieties I") for $D$? The problem is that I do not know the action of the theta-group $$G(D) = \{(P, f) \mid P \in E[2], f\in k(E)^*, \mathrm{div}(f) = 2(P+P_0) - D \}$$ on the vector space $$H^0(E, D) = \{f \in k(E) \mid \mathrm{div}(f) + D \geqslant 0 \} = \langle 1, \frac{1}{x} \rangle.$$
Sorry if this question is very simple, but I did not find an answer in the literature.
