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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 $\, P_1(k) := \gamma^{2k}\lambda^2 + \gamma^{k}\mu^2 + \gamma^{4k}\nu^2\,$ and $\, P_2(k) := P_1(3k).\,$ Let $\,a,b\,$ be any numbers. Define the purported roots of $(2)$ as $\, r_k := a\,P_1(k) + b\,P_2(k).\,$ The polynomial $\, P(z) := (z - r_1)(z - r_2)\cdots (z - r_7)\,$ expands to $\, P(z) = z^7 - 7\, a\, b\,(\lambda^2\, \mu^2 + \lambda^2\, \nu^2 + \mu^2 \, \nu^2)\, z^5 + \dots .$ In equation $(2)$ the $\,z^5\,$ term is missing which implies that $\,a\, b=0\,$ or else $\, \lambda^2\, \mu^2 + \lambda^2\, \nu^2 + \mu^2 \, \nu^2 = 0.\,$ The latter is not true. If $\,b = 0,\,$ then the coefficient of $\, z^3 \,$ is $\, -7\, a^4\, (\lambda^6\, \mu^2 + \mu^6\, \nu^2 + \lambda^2\, \nu^6)\,$ which is not $\,0$. Similarly if $\,a = 0.\,$