On the Lacunarity of some eta-products

The lacunarity is an interesting property of a formal series. We say a series is lacunary if “almost all” of its coefficients are zero. In this article we considered about the lacunarity of some eta-products like \eta(z)^2\eta(bz)^2, and proved that they are lacunary if and only if b is 1,2,3,4 or 16. Then We write them as linear combinations of some CM forms.