TITLE: A 14th and 26th-power Dedekind eta function identity?
QUESTION [28 upvotes]: Given the Dedekind eta function $\eta(\tau)$.  Define $m = (p-1)/2$ and a $24$th root of unity $\zeta = e^{2\pi i/24}$.

Let p be a prime of form $p = 12v+5$. Then for $n = 2,4,8,14$: 
$$\sum_{k=0}^{p-1} \Big(\zeta^{m k}\, \eta\big(\tfrac{\tau+m k}{p}\big)\Big)^n = -\big(\sqrt{p}\;\eta(p\tau)\big)^n\tag1$$
Let p be a prime of form $p = 12v+11$. Then for $n = 2,6,10,14, and\; 26$:
$$\sum_{k=0}^{p-1} \Big(\zeta^{m k}\, \eta\big(\tfrac{\tau+m k}{p}\big)\Big)^n = \big(\sqrt{p}\;\eta(p\tau)\big)^n\tag2$$

Note: One can also use the identity $\zeta^N \eta(\tau) = \eta(\tau+N)$ for a further simplification.
It is easily checked with Mathematica that $(1)$ and $(2)$ hold for hundreds of decimal digits, but are they really true? (Kindly also see this related post.  I have already emailed W. Hart, and he replied he hasn't seen such identities yet.) 
----EDIT----
The $n = 26$ for proposed identity $(2)$ was added courtesy of W.Hart's answer below. (I'm face-palming myself for not checking $n = 26$.)

REPLY [14 votes]: (Intended as a comment to Noam Elkies' response.)
My colleague Marco Streng was kind enough to point out that according to "Eta Products and Theta Series Identities", a book of Guenther Koehler MR2766155: "Serre [128] proved that the Fourier series of a modular form f is lacunary if and only if f is of CM-type, i.e., if f is a linear combination of Hecke theta series. In [129] he showed that eta^r for r=2,4,6,8,10,14,26 are the only even powers of eta which are lacunary."
Here, [129] is J.-P.Serre, Sur la lacunarité des puissances de $\eta$, Glasg. Math.
J, (27) 1985, 203--221.
Note that this is only for even powers of eta, not for odd powers or other kinds of eta product, of which there are numerous lacunary expressions. 
Anyway, this suggests that there ought to be an identity for n = 26, (and possibly others for various eta products?).