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ID: INPUT{id.yaml}
Title: >
$q$-expansion of the Eisenstein series $E_4$
Definition: >
This list contains the coefficients of the
Eisenstein series $E_4(\tau)$ CITE{Wiki}
in its $q$-expansion,
$E_4 = \sum_{n=0}^{\infty} a_n q^n$,
where $q = e^{2\pi i\tau}$.
Parameters:
n:
type: Z
constraints: $n \geq 0$
Comments:
comment-normalization: >
Note that different authors use different normalizations for
Eisenstein series.
Here, we use the normalization that makes
the constant coefficient $a_0 = 1$.
Formulas:
formula-explicit-q-expansion: >
$E_4(\tau) = 1 + 240 \sum_{n=1}^\infty \frac{n^3q^n}{1-q^n}$.
Programs:
program-sage:
language: Sage
code: |
numbers = {n: eisenstein_series_qexp(4,101,
normalization='constant')[n]
for n in [0..100]}
References:
Links:
Wiki:
title: "Wikipedia: Eisenstein series"
url: https://en.wikipedia.org/wiki/Eisenstein_series
Similar tables:
Keywords:
Tags:
- Taylor series
- modular function
- elliptic curves
- number theory
Data properties:
complete: no
Display properties:
number-header: >
$a_n$
Numbers: INPUT{numbers.yaml}
numbers = {n: eisenstein_series_qexp(4,101, normalization='constant')[n] for n in [0..100]}