I. Klein

In "*On the Order-Seven Transformation of Elliptic Functions*" (pp. 287-331), he discusses in p. 298 what we now call the *Klein quartic*,

$$\lambda^3\mu+\mu^3\nu+\nu^3\lambda= 0\tag1$$

and in p. 313 introduces what we can call the *Klein septic resolvent*,

$$z^7-2^2\cdot7^2\big(7\mp\sqrt{-7}\big)z^4+2^5\cdot 7^4\big(5\mp\sqrt{-7}\big)\color{red}z\\ \mp 2^9\cdot3\cdot7^3\sqrt{-7}\frac{g_2}{\sqrt[3]\Delta}=0\tag2$$

(where the red linear $\color{red}z$ is missing in the paper and I assume is a typo). In the same page, he says the roots of $(2)$ in terms of $\lambda,\mu, \nu$ are,

$$z =\frac{\pm 2\sqrt{-7}\Big(P_1+\frac{-1\mp\sqrt{-7}}{2}P_2\Big)}{\sqrt[3]\nabla}\tag3$$

where,

$$P_1 = \gamma^{2x}\lambda^2+\gamma^{x}\mu^2+\gamma^{4x}\nu^2\\ P_2 =\gamma^{6x}\mu\nu+\gamma^{3x}\nu\lambda+\gamma^{5x}\lambda\mu$$

with $\gamma= e^{2\pi i/7}$ as first mentioned in p. 313.

II. Ramanujan

Unbeknownst to Klein (d. 1925), it turns out Ramanujan (d. 1920) found an elegant parameterization to $(1)$. Define the Ramanujan theta function,

$$f(a,b) = \sum_{n=-\infty}^\infty a^{n(n+1)/2}\, b^{n(n-1)/2}$$

then,

$$\lambda = q^{1/56}f(-q^3,-q^4)\\ \mu= \color{red}{-}q^{25/56}f(-q,-q^6)\\ \nu= q^{9/56}f(-q^2,-q^5)\\$$

where $q=e^{2\pi i\tau}$.

III. Question

I tried to implement this in *Mathematica*. Unfortunately, I couldn't get $(3)$ to be a root of the septic $(2)$.

**Q:** Was I wrong in assuming that ** any** parametrization to $(1)$ would do? Or is there some typo or confusion of variables in the paper that screwed up my implementation?