There seems to be a problem with Klein's septic equation $(2)$ combined with the purported roots in $(3)$. Let $\,k\,$ be any integer. Define
$$ P_1(k) := \gamma^{k}\mu^2 + \gamma^{2k}\lambda^2 + \gamma^{4k}\nu^2 \tag{1} $$
and 
$$ P_2(k) := \gamma^{3k}\lambda\nu + \gamma^{6k}\mu\nu +
 \gamma^{5k}\lambda\mu. \tag{2} $$
Let
$$ \,a := 2\sqrt{-7}/\eta(\tau)^2 \; \textrm{ and } \; b := -(7+\sqrt{-7})/\eta(\tau)^2. \tag{3} $$
 Define the roots of $(2)$ as
$$ r_k := a\,P_1(k) + b\,P_2(k). \tag{4} $$
 The polynomial
$\, P(z) := (z - r_1)(z - r_2)\cdots (z - r_7)\,$ expands to 
$$ P(z) = z^7 \!-\! 2^2\! \cdot\! 7^2\, (7+\sqrt{-7})\, z^4 \!+\! 2^5\!\cdot \!7^4\,
 (5+\sqrt{-7})\,z \!+\! 2^7\! \cdot\! 7^3 \sqrt{-7} \frac{g_2(\tau)}{\eta(\tau)^8}. \tag{5}$$
This was for values of $\,\lambda,\mu,\nu\,$ as suggested by Tito Piezas III.

**P.S.** Note  that
$\, \sqrt[3]{\Delta} = \eta(\tau)^8$ is the denominator of the constant term of $P(z).$ Also note that $\, 1728 J(\tau) = j(\tau) = g_2(\tau)^3/\eta(q)^{24}\,$  where $\,J(\tau)\,$ is [Klein's invariant](http://mathworld.wolfram.com/KleinsAbsoluteInvariant.html). Thus using cube roots
$\, 12\sqrt[3]{J(\tau)} = g_2(\tau)/\eta(\tau)^8\,$ and the constant term
can be  written as
$\, 2^9\!\cdot 3 \cdot\! 7^3 \sqrt{-7} \sqrt[3]{J(\tau)} \,$ which is closer to Klein's version.