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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.