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