(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 lacunarite 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?).

(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?).