A Dedekind eta function sum of form $y_0^k+y_1^k+y_2^k+y_3^k+... = 0$

Question

Given the Dedekind eta function $\eta(\tau)$. Define,

$y_p = e^{\pi i p/6}\,\eta(\tfrac{\tau+2p}{5})$

Prove the multi-grade identity [1],

$y_0^k + y_1^k + y_2^k + y_3^k + y_4^k + (\sqrt{5}\,\eta(5\tau))^k = 0$

for k = {2, 4, 8, 14}.

(P.S. This is similar to the Jacobi theta function identity,

$\vartheta_{00}(0;\tau)^4 = \vartheta_{01}(0;\tau)^4 + \vartheta_{10}(0;\tau)^4$

which can also be expressed in terms of the $\eta(\tau)$, though I have no idea how to prove [1].)