Creating spaces of modular forms¶

EXAMPLES:

Sage

sage: m = ModularForms(Gamma1(4),11)
sage: m
Modular Forms space of dimension 6 for
 Congruence Subgroup Gamma1(4) of weight 11 over Rational Field
sage: m.basis()
[
q - 134*q^5 + O(q^6),
q^2 + 80*q^5 + O(q^6),
q^3 + 16*q^5 + O(q^6),
q^4 - 4*q^5 + O(q^6),
1 + 4092/50521*q^2 + 472384/50521*q^3 + 4194300/50521*q^4 + O(q^6),
q + 1024*q^2 + 59048*q^3 + 1048576*q^4 + 9765626*q^5 + O(q^6)
]

Python

>>> from sage.all import *
>>> m = ModularForms(Gamma1(Integer(4)),Integer(11))
>>> m
Modular Forms space of dimension 6 for
 Congruence Subgroup Gamma1(4) of weight 11 over Rational Field
>>> m.basis()
[
q - 134*q^5 + O(q^6),
q^2 + 80*q^5 + O(q^6),
q^3 + 16*q^5 + O(q^6),
q^4 - 4*q^5 + O(q^6),
1 + 4092/50521*q^2 + 472384/50521*q^3 + 4194300/50521*q^4 + O(q^6),
q + 1024*q^2 + 59048*q^3 + 1048576*q^4 + 9765626*q^5 + O(q^6)
]

Sage Live

m = ModularForms(Gamma1(4),11)
m
m.basis()

sage.modular.modform.constructor.CuspForms(group=1, weight=2, base_ring=None, use_cache=True, prec=6)[source]¶

Create a space of cuspidal modular forms.

See the documentation for the ModularForms command for a description of the input parameters.

EXAMPLES:

Sage

sage: CuspForms(11,2)
Cuspidal subspace of dimension 1 of Modular Forms space of dimension 2
 for Congruence Subgroup Gamma0(11) of weight 2 over Rational Field

Python

>>> from sage.all import *
>>> CuspForms(Integer(11),Integer(2))
Cuspidal subspace of dimension 1 of Modular Forms space of dimension 2
 for Congruence Subgroup Gamma0(11) of weight 2 over Rational Field

Sage Live

CuspForms(11,2)

sage.modular.modform.constructor.EisensteinForms(group=1, weight=2, base_ring=None, use_cache=True, prec=6)[source]¶

Create a space of Eisenstein modular forms.

See the documentation for the ModularForms command for a description of the input parameters.

EXAMPLES:

Sage

sage: EisensteinForms(11,2)
Eisenstein subspace of dimension 1 of Modular Forms space of dimension 2
 for Congruence Subgroup Gamma0(11) of weight 2 over Rational Field

Python

>>> from sage.all import *
>>> EisensteinForms(Integer(11),Integer(2))
Eisenstein subspace of dimension 1 of Modular Forms space of dimension 2
 for Congruence Subgroup Gamma0(11) of weight 2 over Rational Field

Sage Live

EisensteinForms(11,2)

sage.modular.modform.constructor.ModularForms(group=1, weight=2, base_ring=None, eis_only=False, use_cache=True, prec=6)[source]¶

Create an ambient space of modular forms.

INPUT:

group – a congruence subgroup or a Dirichlet character eps
weight – integer; the weight (\(\geq 1\))
base_ring – the base ring (ignored if group is a Dirichlet character)
eis_only – if True, compute only the Eisenstein part of the space. Only permitted (and only useful) in weight 1, where computing dimensions of cusp form spaces is expensive.

Create using the command ModularForms(group, weight, base_ring) where group could be either a congruence subgroup or a Dirichlet character.

EXAMPLES: First we create some spaces with trivial character:

Sage

sage: ModularForms(Gamma0(11),2).dimension()
2
sage: ModularForms(Gamma0(1),12).dimension()
2

Python

>>> from sage.all import *
>>> ModularForms(Gamma0(Integer(11)),Integer(2)).dimension()
2
>>> ModularForms(Gamma0(Integer(1)),Integer(12)).dimension()
2

Sage Live

ModularForms(Gamma0(11),2).dimension()
ModularForms(Gamma0(1),12).dimension()

If we give an integer N for the congruence subgroup, it defaults to \(\Gamma_0(N)\):

Sage

sage: ModularForms(1,12).dimension()
2
sage: ModularForms(11,4)
Modular Forms space of dimension 4 for Congruence Subgroup Gamma0(11)
 of weight 4 over Rational Field

Python

>>> from sage.all import *
>>> ModularForms(Integer(1),Integer(12)).dimension()
2
>>> ModularForms(Integer(11),Integer(4))
Modular Forms space of dimension 4 for Congruence Subgroup Gamma0(11)
 of weight 4 over Rational Field

Sage Live

ModularForms(1,12).dimension()
ModularForms(11,4)

We create some spaces for \(\Gamma_1(N)\).

Sage

sage: ModularForms(Gamma1(13),2)
Modular Forms space of dimension 13 for Congruence Subgroup Gamma1(13)
 of weight 2 over Rational Field
sage: ModularForms(Gamma1(13),2).dimension()
13
sage: [ModularForms(Gamma1(7),k).dimension() for k in [2,3,4,5]]
[5, 7, 9, 11]
sage: ModularForms(Gamma1(5),11).dimension()
12

Python

>>> from sage.all import *
>>> ModularForms(Gamma1(Integer(13)),Integer(2))
Modular Forms space of dimension 13 for Congruence Subgroup Gamma1(13)
 of weight 2 over Rational Field
>>> ModularForms(Gamma1(Integer(13)),Integer(2)).dimension()
13
>>> [ModularForms(Gamma1(Integer(7)),k).dimension() for k in [Integer(2),Integer(3),Integer(4),Integer(5)]]
[5, 7, 9, 11]
>>> ModularForms(Gamma1(Integer(5)),Integer(11)).dimension()
12

Sage Live

ModularForms(Gamma1(13),2)
ModularForms(Gamma1(13),2).dimension()
[ModularForms(Gamma1(7),k).dimension() for k in [2,3,4,5]]
ModularForms(Gamma1(5),11).dimension()

We create a space with character:

Sage

sage: # needs sage.rings.number_field
sage: e = (DirichletGroup(13).0)^2
sage: e.order()
6
sage: M = ModularForms(e, 2); M
Modular Forms space of dimension 3, character [zeta6] and weight 2
 over Cyclotomic Field of order 6 and degree 2
sage: f = M.T(2).charpoly('x'); f
x^3 + (-2*zeta6 - 2)*x^2 - 2*zeta6*x + 14*zeta6 - 7
sage: f.factor()
(x - zeta6 - 2) * (x - 2*zeta6 - 1) * (x + zeta6 + 1)

Python

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> e = (DirichletGroup(Integer(13)).gen(0))**Integer(2)
>>> e.order()
6
>>> M = ModularForms(e, Integer(2)); M
Modular Forms space of dimension 3, character [zeta6] and weight 2
 over Cyclotomic Field of order 6 and degree 2
>>> f = M.T(Integer(2)).charpoly('x'); f
x^3 + (-2*zeta6 - 2)*x^2 - 2*zeta6*x + 14*zeta6 - 7
>>> f.factor()
(x - zeta6 - 2) * (x - 2*zeta6 - 1) * (x + zeta6 + 1)

Sage Live

# needs sage.rings.number_field
e = (DirichletGroup(13).0)^2
e.order()
M = ModularForms(e, 2); M
f = M.T(2).charpoly('x'); f
f.factor()

We can also create spaces corresponding to the groups \(\Gamma_H(N)\) intermediate between \(\Gamma_0(N)\) and \(\Gamma_1(N)\):

Sage

sage: G = GammaH(30, [11])
sage: M = ModularForms(G, 2); M
Modular Forms space of dimension 20 for Congruence Subgroup Gamma_H(30)
 with H generated by [11] of weight 2 over Rational Field
sage: M.T(7).charpoly().factor()  # long time (7s on sage.math, 2011)
(x + 4) * x^2 * (x - 6)^4 * (x + 6)^4 * (x - 8)^7 * (x^2 + 4)

Python

>>> from sage.all import *
>>> G = GammaH(Integer(30), [Integer(11)])
>>> M = ModularForms(G, Integer(2)); M
Modular Forms space of dimension 20 for Congruence Subgroup Gamma_H(30)
 with H generated by [11] of weight 2 over Rational Field
>>> M.T(Integer(7)).charpoly().factor()  # long time (7s on sage.math, 2011)
(x + 4) * x^2 * (x - 6)^4 * (x + 6)^4 * (x - 8)^7 * (x^2 + 4)

Sage Live

G = GammaH(30, [11])
M = ModularForms(G, 2); M
M.T(7).charpoly().factor()  # long time (7s on sage.math, 2011)

More examples of spaces with character:

Sage

sage: e = DirichletGroup(5, RationalField()).gen(); e
Dirichlet character modulo 5 of conductor 5 mapping 2 |--> -1

sage: m = ModularForms(e, 2); m
Modular Forms space of dimension 2, character [-1] and weight 2
 over Rational Field
sage: m == loads(dumps(m))
True
sage: m.T(2).charpoly('x')
x^2 - 1
sage: m = ModularForms(e, 6); m.dimension()
4
sage: m.T(2).charpoly('x')
x^4 - 917*x^2 - 42284

Python

>>> from sage.all import *
>>> e = DirichletGroup(Integer(5), RationalField()).gen(); e
Dirichlet character modulo 5 of conductor 5 mapping 2 |--> -1

>>> m = ModularForms(e, Integer(2)); m
Modular Forms space of dimension 2, character [-1] and weight 2
 over Rational Field
>>> m == loads(dumps(m))
True
>>> m.T(Integer(2)).charpoly('x')
x^2 - 1
>>> m = ModularForms(e, Integer(6)); m.dimension()
4
>>> m.T(Integer(2)).charpoly('x')
x^4 - 917*x^2 - 42284

Sage Live

e = DirichletGroup(5, RationalField()).gen(); e
m = ModularForms(e, 2); m
m == loads(dumps(m))
m.T(2).charpoly('x')
m = ModularForms(e, 6); m.dimension()
m.T(2).charpoly('x')

This came up in a subtle bug (Issue #5923):

Sage

sage: ModularForms(gp(1), gap(12))
Modular Forms space of dimension 2 for Modular Group SL(2,Z)
 of weight 12 over Rational Field

Python

>>> from sage.all import *
>>> ModularForms(gp(Integer(1)), gap(Integer(12)))
Modular Forms space of dimension 2 for Modular Group SL(2,Z)
 of weight 12 over Rational Field

Sage Live

ModularForms(gp(1), gap(12))

This came up in another bug (related to Issue #8630):

Sage

sage: chi = DirichletGroup(109, CyclotomicField(3)).0
sage: ModularForms(chi, 2, base_ring = CyclotomicField(15))
Modular Forms space of dimension 10, character [zeta3 + 1] and weight 2
 over Cyclotomic Field of order 15 and degree 8

Python

>>> from sage.all import *
>>> chi = DirichletGroup(Integer(109), CyclotomicField(Integer(3))).gen(0)
>>> ModularForms(chi, Integer(2), base_ring = CyclotomicField(Integer(15)))
Modular Forms space of dimension 10, character [zeta3 + 1] and weight 2
 over Cyclotomic Field of order 15 and degree 8

Sage Live

chi = DirichletGroup(109, CyclotomicField(3)).0
ModularForms(chi, 2, base_ring = CyclotomicField(15))

We create some weight 1 spaces. Here modular symbol algorithms do not work. In some small examples we can prove using Riemann–Roch that there are no cusp forms anyway, so the entire space is Eisenstein:

Sage

sage: M = ModularForms(Gamma1(11), 1); M
Modular Forms space of dimension 5 for Congruence Subgroup Gamma1(11)
 of weight 1 over Rational Field
sage: M.basis()
[
1 + 22*q^5 + O(q^6),
q + 4*q^5 + O(q^6),
q^2 - 4*q^5 + O(q^6),
q^3 - 5*q^5 + O(q^6),
q^4 - 3*q^5 + O(q^6)
]
sage: M.cuspidal_subspace().basis()
[
]
sage: M == M.eisenstein_subspace()
True

Python

>>> from sage.all import *
>>> M = ModularForms(Gamma1(Integer(11)), Integer(1)); M
Modular Forms space of dimension 5 for Congruence Subgroup Gamma1(11)
 of weight 1 over Rational Field
>>> M.basis()
[
1 + 22*q^5 + O(q^6),
q + 4*q^5 + O(q^6),
q^2 - 4*q^5 + O(q^6),
q^3 - 5*q^5 + O(q^6),
q^4 - 3*q^5 + O(q^6)
]
>>> M.cuspidal_subspace().basis()
[
]
>>> M == M.eisenstein_subspace()
True

Sage Live

M = ModularForms(Gamma1(11), 1); M
M.basis()
M.cuspidal_subspace().basis()
M == M.eisenstein_subspace()

When this does not work (which happens as soon as the level is more than about 30), we use the Hecke stability algorithm of George Schaeffer:

Sage

sage: M = ModularForms(Gamma1(57), 1); M  # long time
Modular Forms space of dimension 38 for Congruence Subgroup Gamma1(57)
 of weight 1 over Rational Field
sage: M.cuspidal_submodule().basis()      # long time
[
q - q^4 + O(q^6),
q^3 - q^4 + O(q^6)
]

Python

>>> from sage.all import *
>>> M = ModularForms(Gamma1(Integer(57)), Integer(1)); M  # long time
Modular Forms space of dimension 38 for Congruence Subgroup Gamma1(57)
 of weight 1 over Rational Field
>>> M.cuspidal_submodule().basis()      # long time
[
q - q^4 + O(q^6),
q^3 - q^4 + O(q^6)
]

Sage Live

M = ModularForms(Gamma1(57), 1); M  # long time
M.cuspidal_submodule().basis()      # long time

The Eisenstein subspace in weight 1 can be computed quickly, without triggering the expensive computation of the cuspidal part:

Sage

sage: E = EisensteinForms(Gamma1(59), 1); E  # indirect doctest
Eisenstein subspace of dimension 29 of Modular Forms space for
 Congruence Subgroup Gamma1(59) of weight 1 over Rational Field
sage: (E.0 + E.2).q_expansion(40)
1 + q^2 + 196*q^29 - 197*q^30 - q^31 + q^33 + q^34 + q^37 + q^38 - q^39 + O(q^40)

Python

>>> from sage.all import *
>>> E = EisensteinForms(Gamma1(Integer(59)), Integer(1)); E  # indirect doctest
Eisenstein subspace of dimension 29 of Modular Forms space for
 Congruence Subgroup Gamma1(59) of weight 1 over Rational Field
>>> (E.gen(0) + E.gen(2)).q_expansion(Integer(40))
1 + q^2 + 196*q^29 - 197*q^30 - q^31 + q^33 + q^34 + q^37 + q^38 - q^39 + O(q^40)

Sage Live

E = EisensteinForms(Gamma1(59), 1); E  # indirect doctest
(E.0 + E.2).q_expansion(40)

sage.modular.modform.constructor.ModularForms_clear_cache()[source]¶

Clear the cache of modular forms.

EXAMPLES:

Sage

sage: M = ModularForms(37,2)
sage: sage.modular.modform.constructor._cache == {}
False

Python

>>> from sage.all import *
>>> M = ModularForms(Integer(37),Integer(2))
>>> sage.modular.modform.constructor._cache == {}
False

Sage Live

M = ModularForms(37,2)
sage.modular.modform.constructor._cache == {}

Sage

sage: sage.modular.modform.constructor.ModularForms_clear_cache()
sage: sage.modular.modform.constructor._cache
{}

Python

>>> from sage.all import *
>>> sage.modular.modform.constructor.ModularForms_clear_cache()
>>> sage.modular.modform.constructor._cache
{}

Sage Live

sage.modular.modform.constructor.ModularForms_clear_cache()
sage.modular.modform.constructor._cache

Python

>>> from sage.all import *
>>> sage.modular.modform.constructor.ModularForms_clear_cache()
>>> sage.modular.modform.constructor._cache
{}

Sage Live

sage.modular.modform.constructor.ModularForms_clear_cache()
sage.modular.modform.constructor._cache

sage.modular.modform.constructor.Newform(identifier, group=None, weight=2, base_ring=Rational Field, names=None)[source]¶

INPUT:

identifier – a canonical label, or the index of the specific newform desired
group – the congruence subgroup of the newform
weight – the weight of the newform (default: 2)
base_ring – the base ring
names – if the newform has coefficients in a number field, a generator name must be specified

EXAMPLES:

Sage

sage: Newform('67a', names='a')
q + 2*q^2 - 2*q^3 + 2*q^4 + 2*q^5 + O(q^6)
sage: Newform('67b', names='a')
q + a1*q^2 + (-a1 - 3)*q^3 + (-3*a1 - 3)*q^4 - 3*q^5 + O(q^6)

Python

>>> from sage.all import *
>>> Newform('67a', names='a')
q + 2*q^2 - 2*q^3 + 2*q^4 + 2*q^5 + O(q^6)
>>> Newform('67b', names='a')
q + a1*q^2 + (-a1 - 3)*q^3 + (-3*a1 - 3)*q^4 - 3*q^5 + O(q^6)

Sage Live

Newform('67a', names='a')
Newform('67b', names='a')

sage.modular.modform.constructor.Newforms(group, weight=2, base_ring=None, names=None)[source]¶

Return a list of the newforms of the given weight and level (or weight, level and character). These are calculated as \(\operatorname{Gal}(\overline{F} / F)\)-orbits, where \(F\) is the given base field.

INPUT:

group – the congruence subgroup of the newform, or a Nebentypus character
weight – the weight of the newform (default: 2)
base_ring – the base ring (defaults to \(\QQ\) for spaces without character, or the base ring of the character otherwise)
names – if the newform has coefficients in a number field, a generator name must be specified

EXAMPLES:

Sage

sage: Newforms(11, 2)
[q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6)]
sage: Newforms(65, names='a')
[q - q^2 - 2*q^3 - q^4 - q^5 + O(q^6),
 q + a1*q^2 + (a1 + 1)*q^3 + (-2*a1 - 1)*q^4 + q^5 + O(q^6),
 q + a2*q^2 + (-a2 + 1)*q^3 + q^4 - q^5 + O(q^6)]

Python

>>> from sage.all import *
>>> Newforms(Integer(11), Integer(2))
[q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6)]
>>> Newforms(Integer(65), names='a')
[q - q^2 - 2*q^3 - q^4 - q^5 + O(q^6),
 q + a1*q^2 + (a1 + 1)*q^3 + (-2*a1 - 1)*q^4 + q^5 + O(q^6),
 q + a2*q^2 + (-a2 + 1)*q^3 + q^4 - q^5 + O(q^6)]

Sage Live

Newforms(11, 2)
Newforms(65, names='a')

A more complicated example involving both a nontrivial character, and a base field that is not minimal for that character:

Sage

sage: K.<i> = QuadraticField(-1)
sage: chi = DirichletGroup(5, K)[1]
sage: len(Newforms(chi, 7, names='a'))
1
sage: x = polygen(K); L.<c> = K.extension(x^2 - 402*i)
sage: N = Newforms(chi, 7, base_ring = L); len(N)
2
sage: sorted([N[0][2], N[1][2]]) == sorted([1/2*c - 5/2*i - 5/2, -1/2*c - 5/2*i - 5/2])
True

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> chi = DirichletGroup(Integer(5), K)[Integer(1)]
>>> len(Newforms(chi, Integer(7), names='a'))
1
>>> x = polygen(K); L = K.extension(x**Integer(2) - Integer(402)*i, names=('c',)); (c,) = L._first_ngens(1)
>>> N = Newforms(chi, Integer(7), base_ring = L); len(N)
2
>>> sorted([N[Integer(0)][Integer(2)], N[Integer(1)][Integer(2)]]) == sorted([Integer(1)/Integer(2)*c - Integer(5)/Integer(2)*i - Integer(5)/Integer(2), -Integer(1)/Integer(2)*c - Integer(5)/Integer(2)*i - Integer(5)/Integer(2)])
True

Sage Live

K.<i> = QuadraticField(-1)
chi = DirichletGroup(5, K)[1]
len(Newforms(chi, 7, names='a'))
x = polygen(K); L.<c> = K.extension(x^2 - 402*i)
N = Newforms(chi, 7, base_ring = L); len(N)
sorted([N[0][2], N[1][2]]) == sorted([1/2*c - 5/2*i - 5/2, -1/2*c - 5/2*i - 5/2])

sage.modular.modform.constructor.canonical_parameters(group, level, weight, base_ring)[source]¶

Given a group, level, weight, and base_ring as input by the user, return a canonicalized version of them, where level is a Sage integer, group really is a group, weight is a Sage integer, and base_ring a Sage ring. Note that we can’t just get the level from the group, because we have the convention that the character for Gamma1(N) is None (which makes good sense).

INPUT:

group – integer, group, or Dirichlet character
level – integer or group
weight – coercible to integer
base_ring – commutative ring

OUTPUT:

level – integer
group – congruence subgroup
weight – integer
ring – commutative ring

EXAMPLES:

Sage

sage: from sage.modular.modform.constructor import canonical_parameters
sage: v = canonical_parameters(5, 5, int(7), ZZ); v
(5, Congruence Subgroup Gamma0(5), 7, Integer Ring)
sage: type(v[0]), type(v[1]), type(v[2]), type(v[3])
(<class 'sage.rings.integer.Integer'>,
 <class 'sage.modular.arithgroup.congroup_gamma0.Gamma0_class_with_category'>,
 <class 'sage.rings.integer.Integer'>,
 <class 'sage.rings.integer_ring.IntegerRing_class'>)
sage: canonical_parameters( 5, 7, 7, ZZ )
Traceback (most recent call last):
...
ValueError: group and level do not match.

Python

>>> from sage.all import *
>>> from sage.modular.modform.constructor import canonical_parameters
>>> v = canonical_parameters(Integer(5), Integer(5), int(Integer(7)), ZZ); v
(5, Congruence Subgroup Gamma0(5), 7, Integer Ring)
>>> type(v[Integer(0)]), type(v[Integer(1)]), type(v[Integer(2)]), type(v[Integer(3)])
(<class 'sage.rings.integer.Integer'>,
 <class 'sage.modular.arithgroup.congroup_gamma0.Gamma0_class_with_category'>,
 <class 'sage.rings.integer.Integer'>,
 <class 'sage.rings.integer_ring.IntegerRing_class'>)
>>> canonical_parameters( Integer(5), Integer(7), Integer(7), ZZ )
Traceback (most recent call last):
...
ValueError: group and level do not match.

Sage Live

from sage.modular.modform.constructor import canonical_parameters
v = canonical_parameters(5, 5, int(7), ZZ); v
type(v[0]), type(v[1]), type(v[2]), type(v[3])
canonical_parameters( 5, 7, 7, ZZ )

sage.modular.modform.constructor.parse_label(s)[source]¶

Given a string s corresponding to a newform label, return the corresponding group and index.

EXAMPLES:

Sage

sage: sage.modular.modform.constructor.parse_label('11a')
(Congruence Subgroup Gamma0(11), 0)
sage: sage.modular.modform.constructor.parse_label('11aG1')
(Congruence Subgroup Gamma1(11), 0)
sage: sage.modular.modform.constructor.parse_label('11wG1')
(Congruence Subgroup Gamma1(11), 22)

Python

>>> from sage.all import *
>>> sage.modular.modform.constructor.parse_label('11a')
(Congruence Subgroup Gamma0(11), 0)
>>> sage.modular.modform.constructor.parse_label('11aG1')
(Congruence Subgroup Gamma1(11), 0)
>>> sage.modular.modform.constructor.parse_label('11wG1')
(Congruence Subgroup Gamma1(11), 22)

Sage Live

sage.modular.modform.constructor.parse_label('11a')
sage.modular.modform.constructor.parse_label('11aG1')
sage.modular.modform.constructor.parse_label('11wG1')

GammaH labels should also return the group and index (Issue #20823):

Sage

sage: sage.modular.modform.constructor.parse_label('389cGH[16]')
(Congruence Subgroup Gamma_H(389) with H generated by [16], 2)

Python

>>> from sage.all import *
>>> sage.modular.modform.constructor.parse_label('389cGH[16]')
(Congruence Subgroup Gamma_H(389) with H generated by [16], 2)

Sage Live

sage.modular.modform.constructor.parse_label('389cGH[16]')