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ID: INPUT{id.yaml}

Title: >
  $q$-expansion of the modular discriminant $\Delta$

Definition: >
  This list contains the coefficients of the 
  modular discrimiant $\Delta$ CITE{Wiki}
  in its $q$-expansion, 
  $\Delta = \sum_{n=0}^{\infty} \tau(n) q^n$.
  The coefficients are the values of the Ramanujan tau-function.
  
Parameters:
  n:
    type: Z
    constraints: $n \geq 0$
  
Comments:
  comment-normalization: >
    Note that different authors use different normalization factors for
    $\Delta$ CITE{Wiki}, CITE{WikiTau}.
    Here, we use the normalization that 
    makes $\Delta$ a normalized cusp form (of weight 12 and level 1).

Formulas:

Programs: 
  program-sage:
    language: Sage
    code: |
      numbers = {n: delta_qexp(101)[n] for n in [0..100]}

References:

Links:
  Wiki:
    title: "Wikipedia: Modular discriminant"
    url: https://en.wikipedia.org/wiki/Weierstrass%27s_elliptic_functions#Modular_discriminant
  WikiTau:
    title: "Wikipedia: Ramanujan tau function"
    url: https://en.wikipedia.org/wiki/Ramanujan_tau_function
    
Similar tables:
  
Keywords:
  
Tags:
- Taylor series
- modular function
- elliptic curves
- number theory

Data properties:
  type: Z
  complete: no
  
Display properties:
  number-header: >
    $\tau(n)$

Numbers: INPUT{numbers.yaml}

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$q$-expansion of the modular discriminant $\Delta$

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Numbers

$n$

$\tau(n)$

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

Definition

This list contains the coefficients of the modular discrimiant $\Delta$ [1] in its $q$-expansion, $\Delta = \sum_{n=0}^{\infty} \tau(n) q^n$. The coefficients are the values of the Ramanujan tau-function.

Parameters

$n$

— integer ($n \geq 0$)

Comments

(1)

Note that different authors use different normalization factors for $\Delta$ [1], [2]. Here, we use the normalization that makes $\Delta$ a normalized cusp form (of weight 12 and level 1).

Programs

(P1)

Sage

numbers = {n: delta_qexp(101)[n] for n in [0..100]}

Links

[1]

Wikipedia: Modular discriminant

[2]

Wikipedia: Ramanujan tau function

Data properties

Entries are of type: integer

Table is complete: no