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MA342J: Introduction to Modular Forms Homework 3, due on February 16 In class we proved that every modular form can be written as a polynomial in the Eisenstein series E4 = 1 + 240 E6 = 1 − 504 ∞ X n=1 ∞ X σ3 (n)q n σ5 (n)q n n=1 Moreover, since there are no polynomial relations between E4 and E6 this representation is unique. 1. Write forms a) E14 = 1 − 24 ∞ X σ13 (n)q n n=1 b) E18 = 1 − c) ∞ 28728 X σ17 (n)q n 43867 n=1 cusp form f = q + 216 q 2 + ... of weight 16 as polynomials in E4 and E6 . 2. Fourier expansion of a form of weight 16 starts as 2 + 192q + . . . Find the Fourier coefficient of this form near q 2 . 1