The Victor Miller basis - Modular Forms

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# The Victor Miller basis

This module contains functions for quick calculation of a basis of \(q\)-expansions for the space of modular forms of level 1 and any weight. The basis returned is the Victor Miller basis, which is the unique basis of elliptic modular forms \(f_1, \dots, f_d\) for which \(a_i(f_j) = \delta_{ij}\) for \(1 \le i, j \le d\) (where \(d\) is the dimension of the space).

This basis is calculated using a standard set of generators for the ring of modular forms, using the fast multiplication algorithms for polynomials and power series provided by the FLINT library. (This is far quicker than using modular symbols).

sage.modular.modform.vm_basis.delta_qexp(prec=10, var='q', K=Integer Ring)

  Return the \(q\)-expansion of the weight 12 cusp form \(\Delta\) as a power series with coefficients in the ring K (\(= \ZZ\) by default).

  INPUT:

  - `prec` – integer (default: 10); the absolute precision of the output (must be positive)

  - `var` – string (default: `'q'`); variable name

  - `K` – ring (default: \(\ZZ\)); base ring of answer

  OUTPUT: a power series over K in the variable `var`

  ALGORITHM:

  Compute the theta series

  \[\sum_{n \ge 0} (-1)^n (2n+1) q^{n(n+1)/2},\]

  a very simple explicit modular form whose 8th power is \(\Delta\). Then compute the 8th power. All computations are done over \(\ZZ\) or \(\ZZ\) modulo \(N\) depending on the characteristic of the given coefficient ring \(K\), and coerced into \(K\) afterwards.

  EXAMPLES:

  Sage

      sage: delta_qexp(7)
      q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 + O(q^7)
      sage: delta_qexp(7,'z')
      z - 24*z^2 + 252*z^3 - 1472*z^4 + 4830*z^5 - 6048*z^6 + O(z^7)
      sage: delta_qexp(-3)
      Traceback (most recent call last):
      ...
      ValueError: prec must be positive
      sage: delta_qexp(20, K = GF(3))
      q + q^4 + 2*q^7 + 2*q^13 + q^16 + 2*q^19 + O(q^20)
      sage: delta_qexp(20, K = GF(3^5, 'a'))
      q + q^4 + 2*q^7 + 2*q^13 + q^16 + 2*q^19 + O(q^20)
      sage: delta_qexp(10, K = IntegerModRing(60))
      q + 36*q^2 + 12*q^3 + 28*q^4 + 30*q^5 + 12*q^6 + 56*q^7 + 57*q^9 + O(q^10)

  Python

      >>> from sage.all import *
      >>> delta_qexp(Integer(7))
      q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 + O(q^7)
      >>> delta_qexp(Integer(7),'z')
      z - 24*z^2 + 252*z^3 - 1472*z^4 + 4830*z^5 - 6048*z^6 + O(z^7)
      >>> delta_qexp(-Integer(3))
      Traceback (most recent call last):
      ...
      ValueError: prec must be positive
      >>> delta_qexp(Integer(20), K = GF(Integer(3)))
      q + q^4 + 2*q^7 + 2*q^13 + q^16 + 2*q^19 + O(q^20)
      >>> delta_qexp(Integer(20), K = GF(Integer(3)**Integer(5), 'a'))
      q + q^4 + 2*q^7 + 2*q^13 + q^16 + 2*q^19 + O(q^20)
      >>> delta_qexp(Integer(10), K = IntegerModRing(Integer(60)))
      q + 36*q^2 + 12*q^3 + 28*q^4 + 30*q^5 + 12*q^6 + 56*q^7 + 57*q^9 + O(q^10)

  AUTHORS:

  - William Stein: original code

  - David Harvey (2007-05): sped up first squaring step

  - Martin Raum (2009-08-02): use FLINT for polynomial arithmetic (instead of NTL)

sage.modular.modform.vm_basis.victor_miller_basis(k, prec=10, cusp_only=False, var='q')

  Compute and return the Victor Miller basis for modular forms of weight \(k\) and level 1 to precision \(O(q^{prec})\). If `cusp_only` is True, return only a basis for the cuspidal subspace.

  INPUT:

  - `k` – integer

  - `prec` – (default: 10) a positive integer

  - `cusp_only` – boolean (default: `False`)

  - `var` – string (default: `'q'`)

  OUTPUT: a sequence whose entries are power series in `ZZ[[var]]`

  EXAMPLES:

  Sage

      sage: victor_miller_basis(1, 6)
      []
      sage: victor_miller_basis(0, 6)
      [1 + O(q^6)]
      sage: victor_miller_basis(2, 6)
      []
      sage: victor_miller_basis(4, 6)
      [1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + O(q^6)]

      sage: victor_miller_basis(6, 6, var='w')
      [1 - 504*w - 16632*w^2 - 122976*w^3 - 532728*w^4 - 1575504*w^5 + O(w^6)]

      sage: victor_miller_basis(6, 6)
      [1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 + O(q^6)]
      sage: victor_miller_basis(12, 6)
      [1 + 196560*q^2 + 16773120*q^3 + 398034000*q^4 + 4629381120*q^5 + O(q^6),
       q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)]

      sage: victor_miller_basis(12, 6, cusp_only=True)
      [q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)]
      sage: victor_miller_basis(24, 6, cusp_only=True)
      [q + 195660*q^3 + 12080128*q^4 + 44656110*q^5 + O(q^6),
       q^2 - 48*q^3 + 1080*q^4 - 15040*q^5 + O(q^6)]
      sage: victor_miller_basis(24, 6)
      [1 + 52416000*q^3 + 39007332000*q^4 + 6609020221440*q^5 + O(q^6),
       q + 195660*q^3 + 12080128*q^4 + 44656110*q^5 + O(q^6),
       q^2 - 48*q^3 + 1080*q^4 - 15040*q^5 + O(q^6)]
      sage: victor_miller_basis(32, 6)
      [1 + 2611200*q^3 + 19524758400*q^4 + 19715347537920*q^5 + O(q^6),
       q + 50220*q^3 + 87866368*q^4 + 18647219790*q^5 + O(q^6),
       q^2 + 432*q^3 + 39960*q^4 - 1418560*q^5 + O(q^6)]

      sage: victor_miller_basis(40,200)[1:] == victor_miller_basis(40,200,cusp_only=True)
      True
      sage: victor_miller_basis(200,40)[1:] == victor_miller_basis(200,40,cusp_only=True)
      True

  Python

      >>> from sage.all import *
      >>> victor_miller_basis(Integer(1), Integer(6))
      []
      >>> victor_miller_basis(Integer(0), Integer(6))
      [1 + O(q^6)]
      >>> victor_miller_basis(Integer(2), Integer(6))
      []
      >>> victor_miller_basis(Integer(4), Integer(6))
      [1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + O(q^6)]

      >>> victor_miller_basis(Integer(6), Integer(6), var='w')
      [1 - 504*w - 16632*w^2 - 122976*w^3 - 532728*w^4 - 1575504*w^5 + O(w^6)]

      >>> victor_miller_basis(Integer(6), Integer(6))
      [1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 + O(q^6)]
      >>> victor_miller_basis(Integer(12), Integer(6))
      [1 + 196560*q^2 + 16773120*q^3 + 398034000*q^4 + 4629381120*q^5 + O(q^6),
       q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)]

      >>> victor_miller_basis(Integer(12), Integer(6), cusp_only=True)
      [q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)]
      >>> victor_miller_basis(Integer(24), Integer(6), cusp_only=True)
      [q + 195660*q^3 + 12080128*q^4 + 44656110*q^5 + O(q^6),
       q^2 - 48*q^3 + 1080*q^4 - 15040*q^5 + O(q^6)]
      >>> victor_miller_basis(Integer(24), Integer(6))
      [1 + 52416000*q^3 + 39007332000*q^4 + 6609020221440*q^5 + O(q^6),
       q + 195660*q^3 + 12080128*q^4 + 44656110*q^5 + O(q^6),
       q^2 - 48*q^3 + 1080*q^4 - 15040*q^5 + O(q^6)]
      >>> victor_miller_basis(Integer(32), Integer(6))
      [1 + 2611200*q^3 + 19524758400*q^4 + 19715347537920*q^5 + O(q^6),
       q + 50220*q^3 + 87866368*q^4 + 18647219790*q^5 + O(q^6),
       q^2 + 432*q^3 + 39960*q^4 - 1418560*q^5 + O(q^6)]

      >>> victor_miller_basis(Integer(40),Integer(200))[Integer(1):] == victor_miller_basis(Integer(40),Integer(200),cusp_only=True)
      True
      >>> victor_miller_basis(Integer(200),Integer(40))[Integer(1):] == victor_miller_basis(Integer(200),Integer(40),cusp_only=True)
      True

  AUTHORS:

  - William Stein, Craig Citro: original code

  - Martin Raum (2009-08-02): use FLINT for polynomial arithmetic (instead of NTL)