The Victor Miller Basis

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

Return the q-expansion of Delta as a power series with coefficients in K (=ZZ by default).

INPUT:

  • prec – integer; the absolute precision of the output
  • var – (default: ‘q’) variable name
  • K – (default: ZZ) base ring of answer
OUTPUT:
a power series over K
ALGORITHM:
Compute a simple very explicit modular form whose 8th power is Delta. Then compute the 8th power using NTL polynomial arithmetic, which is VERY fast. This function computes a million terms of Delta in under a minute.

EXAMPLES:

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)
...
ValueError: prec must be positive

AUTHORS:

  • William Stein: original code
  • David Harvey (2007-05): sped up first squaring step
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 – an integer
  • prec – (default: 10) a positive integer
  • cusp_only – bool (default: False)
  • var – string (default: ‘q’
OUTPUT:
A sequence whose entries are power series in ZZ[[var]].

EXAMPLES:

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

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