Supplement material to an article (RISC report 25-01) by Ralf Hemmecke, Peter Paule, and Silviu Radu.
We start with a modular form $$g(\tau):=(1+\lambda(\tau))\theta_3(\tau)^4 \in M_2(\Gamma(2))$$ and a modular function $$h(\tau) := \frac{\lambda(\tau) (1-\lambda(\tau))^2}{16(1+\lambda(\tau))^4} \in M_0(\Gamma(2))$$ where $\lambda(\tau)$ is the modular lambda function and $\theta_3(\tau)=\theta_3(0,\tau)$ is the Jacobi theta function.
Then define $$\iota(\tau) := \frac{1}{h(\tau)}$$ and $$K(\tau) := \frac{29}{24}\iota(\tau)\iota(N\tau) \frac{H(\tau)}{g(\tau)}$$ where $$G(\tau) := \frac{w}{2 \pi i}\, \frac{g'(\tau)}{g(\tau)}\qquad \text{ with }w=2$$ and $$H(\tau):=G(\tau)- N\, G(N \tau).$$
The following data was computed by the package QEta.