Given the Dedekind eta function $\eta(\tau)$ and complex numbers a,b with imaginary part > 0, anybody knows how to prove the proposed identity,
$$\sum_{k=0}^{p-1} e^{2\pi i k/4}\eta^3\big(\tfrac{a+k}{p}\big)\eta^3\big(\tfrac{b+k}{p}\big) = p^3\eta^3(p a)\eta^3(p b)$$
where p is ANY prime of form $p = 4n-1$.
(This was inspired by Berndt and Hart's paper "An Identity for the Dedekind eta function involving two independent complex variables" (2007) wherein they discussed the case p = 3 but not p = 7, 11, etc.)
