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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:
  type: Z
  complete: no
  
Display properties:
  number-header: >
    $a_n$

Numbers: INPUT{numbers.yaml}

xxxxxxxxxx
 
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:
  type: Z
  complete: no
  
Display properties:
  number-header: >
    $a_n$
​
Numbers: INPUT{numbers.yaml}
​
​

— preview —

q

-expansion of the Eisenstein series

E_{4}

Numbers

n

a_{n}

INPUT{numbers.yaml} (not shown in preview)

Definition

This list contains the coefficients of the Eisenstein series

E_{4} (τ)

[1] in its

q

-expansion,

E_{4} = \sum_{n = 0}^{\infty} a_{n} q^{n}

, where

q = e^{2 π i τ}

.

Parameters

n

— integer (

n \geq 0

)

Formulas

(1)

E_{4} (τ) = 1 + 240 \sum_{n = 1}^{\infty} \frac{n^{3} q^{n}}{1 - q^{n}}

.

Comments

(2)

Note that different authors use different normalizations for Eisenstein series. Here, we use the normalization that makes the constant coefficient

a_{0} = 1

.

Programs

(P1)

Sage

numbers = {n: eisenstein_series_qexp(4,101,
                  normalization='constant')[n] 
              for n in [0..100]}

Links

[1]

Wikipedia: Eisenstein series

Data properties

Entries are of type: integer

Table is complete: no