On representation of modular forms as homogeneous polynomials

In the article we study the spaces of modular forms such that each element of them is a homogeneous polynomial of modular forms of low weights of the same level. It is a classical fact that it is true for the level 1. N. Koblitz point out that it is true for cusp forms of level 4. In this article we show that the analogous situation takes place for the levels corresponding to the eta-products with multiplicative coefficients. In all cases under consideration the base functions are eta-products. In each case the base functions are written explicitly. Dimensions of spaces are calculated by the Cohen–Oesterle formula, the orders in cusps are calculated by the Biagioli formula.