On Monomial Relations Between Eisenstein Series

We show that the relations E₈ = E²₄, E₁₀ =E₄E₆, and E₁₄ = E₆ E₈ are the only relations of the form E_k = E_l E_k-l , where E_k is the Eisen-stein series of weight k for the full modular group. More generally we show that, apart from fifteen exceptions, the product of two level one eigenforms is never an eigenform. The proof uses the Rankin-Selberg method.