The 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 
$$ P(z) = z^7 - 2^2\cdot 7^2\, (7+\sqrt{-7})\, z^4 + 2^5\cdot 7^4\,
 (5+\sqrt{-7})\,  z + 2^7 \cdot 7^3 \sqrt{-7} g_2/\eta(\tau)^8. \tag{5}$$
This was for values of $\,\lambda,\mu,\nu\,$ as suggested by Tito Piezas III.