Creation of modular symbols spaces¶
EXAMPLES: We create a space and output its category.
sage: C = HeckeModules(RationalField()); C
Category of Hecke modules over Rational Field
sage: M = ModularSymbols(11)
sage: M.category()
Category of Hecke modules over Rational Field
sage: M in C
True
We create a space compute the charpoly, then compute the same but over a bigger field. In each case we also decompose the space using \(T_2\).
sage: M = ModularSymbols(23,2,base_ring=QQ)
sage: M.T(2).charpoly('x').factor()
(x - 3) * (x^2 + x - 1)^2
sage: M.decomposition(2)
[
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 5 for Gamma_0(23) of weight 2 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 5 for Gamma_0(23) of weight 2 with sign 0 over Rational Field
]
sage: M = ModularSymbols(23,2,base_ring=QuadraticField(5, 'sqrt5'))
sage: M.T(2).charpoly('x').factor()
(x - 3) * (x - 1/2*sqrt5 + 1/2)^2 * (x + 1/2*sqrt5 + 1/2)^2
sage: M.decomposition(2)
[
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 5 for Gamma_0(23) of weight 2 with sign 0 over Number Field in sqrt5 with defining polynomial x^2 - 5 with sqrt5 = 2.236067977499790?,
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 5 for Gamma_0(23) of weight 2 with sign 0 over Number Field in sqrt5 with defining polynomial x^2 - 5 with sqrt5 = 2.236067977499790?,
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 5 for Gamma_0(23) of weight 2 with sign 0 over Number Field in sqrt5 with defining polynomial x^2 - 5 with sqrt5 = 2.236067977499790?
]
We compute some Hecke operators and do a consistency check:
sage: m = ModularSymbols(39, 2)
sage: t2 = m.T(2); t5 = m.T(5)
sage: t2*t5 - t5*t2 == 0
True
This tests the bug reported in trac ticket #1220:
sage: G = GammaH(36, [13, 19])
sage: G.modular_symbols()
Modular Symbols space of dimension 13 for Congruence Subgroup Gamma_H(36) with H generated by [13, 19] of weight 2 with sign 0 and over Rational Field
sage: G.modular_symbols().cuspidal_subspace()
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 13 for Congruence Subgroup Gamma_H(36) with H generated by [13, 19] of weight 2 with sign 0 and over Rational Field
This test catches a tricky corner case for spaces with character:
sage: ModularSymbols(DirichletGroup(20).1**3, weight=3, sign=1).cuspidal_subspace()
Modular Symbols subspace of dimension 3 of Modular Symbols space of dimension 6 and level 20, weight 3, character [1, -zeta4], sign 1, over Cyclotomic Field of order 4 and degree 2
This tests the bugs reported in trac ticket #20932:
sage: chi = kronecker_character(3*34603)
sage: ModularSymbols(chi, 2, sign=1, base_ring=GF(3)) # not tested # long time (600 seconds)
Modular Symbols space of dimension 11535 and level 103809, weight 2, character [2, 2], sign 1, over Finite Field of size 3
sage: chi = kronecker_character(3*61379)
sage: ModularSymbols(chi, 2, sign=1, base_ring=GF(3)) # not tested # long time (1800 seconds)
Modular Symbols space of dimension 20460 and level 184137, weight 2, character [2, 2], sign 1, over Finite Field of size 3
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sage.modular.modsym.modsym.ModularSymbols(group=1, weight=2, sign=0, base_ring=None, use_cache=True, custom_init=None)¶ Create an ambient space of modular symbols.
INPUT:
group- A congruence subgroup or a Dirichlet character eps.weight- int, the weight, which must be >= 2.sign- int, The sign of the involution on modular symbols induced by complex conjugation. The default is 0, which means “no sign”, i.e., take the whole space.base_ring- the base ring. Defaults to \(\QQ\) if no character is given, or to the minimal extension of \(\QQ\) containing the values of the character.custom_init- a function that is called with self as input before any computations are done using self; this could be used to set a custom modular symbols presentation. If self is already in the cache and use_cache=True, then this function is not called.
EXAMPLES: First we create some spaces with trivial character:
sage: ModularSymbols(Gamma0(11),2).dimension() 3 sage: ModularSymbols(Gamma0(1),12).dimension() 3
If we give an integer N for the congruence subgroup, it defaults to \(\Gamma_0(N)\):
sage: ModularSymbols(1,12,-1).dimension() 1 sage: ModularSymbols(11,4, sign=1) Modular Symbols space of dimension 4 for Gamma_0(11) of weight 4 with sign 1 over Rational Field
We create some spaces for \(\Gamma_1(N)\).
sage: ModularSymbols(Gamma1(13),2) Modular Symbols space of dimension 15 for Gamma_1(13) of weight 2 with sign 0 and over Rational Field sage: ModularSymbols(Gamma1(13),2, sign=1).dimension() 13 sage: ModularSymbols(Gamma1(13),2, sign=-1).dimension() 2 sage: [ModularSymbols(Gamma1(7),k).dimension() for k in [2,3,4,5]] [5, 8, 12, 16] sage: ModularSymbols(Gamma1(5),11).dimension() 20
We create a space for \(\Gamma_H(N)\):
sage: G = GammaH(15,[4,13]) sage: M = ModularSymbols(G,2) sage: M.decomposition() [ Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 5 for Congruence Subgroup Gamma_H(15) with H generated by [4, 7] of weight 2 with sign 0 and over Rational Field, Modular Symbols subspace of dimension 3 of Modular Symbols space of dimension 5 for Congruence Subgroup Gamma_H(15) with H generated by [4, 7] of weight 2 with sign 0 and over Rational Field ]
We create a space with character:
sage: e = (DirichletGroup(13).0)^2 sage: e.order() 6 sage: M = ModularSymbols(e, 2); M Modular Symbols space of dimension 4 and level 13, weight 2, character [zeta6], sign 0, over Cyclotomic Field of order 6 and degree 2 sage: f = M.T(2).charpoly('x'); f x^4 + (-zeta6 - 1)*x^3 - 8*zeta6*x^2 + (10*zeta6 - 5)*x + 21*zeta6 - 21 sage: f.factor() (x - zeta6 - 2) * (x - 2*zeta6 - 1) * (x + zeta6 + 1)^2
We create a space with character over a larger base ring than the values of the character:
sage: ModularSymbols(e, 2, base_ring = CyclotomicField(24)) Modular Symbols space of dimension 4 and level 13, weight 2, character [zeta24^4], sign 0, over Cyclotomic Field of order 24 and degree 8
More examples of spaces with character:
sage: e = DirichletGroup(5, RationalField()).gen(); e Dirichlet character modulo 5 of conductor 5 mapping 2 |--> -1 sage: m = ModularSymbols(e, 2); m Modular Symbols space of dimension 2 and level 5, weight 2, character [-1], sign 0, over Rational Field
sage: m.T(2).charpoly('x') x^2 - 1 sage: m = ModularSymbols(e, 6); m.dimension() 6 sage: m.T(2).charpoly('x') x^6 - 873*x^4 - 82632*x^2 - 1860496
We create a space of modular symbols with nontrivial character in characteristic 2.
sage: G = DirichletGroup(13,GF(4,'a')); G Group of Dirichlet characters modulo 13 with values in Finite Field in a of size 2^2 sage: e = G.list()[2]; e Dirichlet character modulo 13 of conductor 13 mapping 2 |--> a + 1 sage: M = ModularSymbols(e,4); M Modular Symbols space of dimension 8 and level 13, weight 4, character [a + 1], sign 0, over Finite Field in a of size 2^2 sage: M.basis() ([X*Y,(1,0)], [X*Y,(1,5)], [X*Y,(1,10)], [X*Y,(1,11)], [X^2,(0,1)], [X^2,(1,10)], [X^2,(1,11)], [X^2,(1,12)]) sage: M.T(2).matrix() [ 0 0 0 0 0 0 1 1] [ 0 0 0 0 0 0 0 0] [ 0 0 0 0 0 a + 1 1 a] [ 0 0 0 0 0 1 a + 1 a] [ 0 0 0 0 a + 1 0 1 1] [ 0 0 0 0 0 a 1 a] [ 0 0 0 0 0 0 a + 1 a] [ 0 0 0 0 0 0 1 0]
We illustrate the custom_init function, which can be used to make arbitrary changes to the modular symbols object before its presentation is computed:
sage: ModularSymbols_clear_cache() sage: def custom_init(M): ....: M.customize='hi' sage: M = ModularSymbols(1,12, custom_init=custom_init) sage: M.customize 'hi'
We illustrate the relation between custom_init and use_cache:
sage: def custom_init(M): ....: M.customize='hi2' sage: M = ModularSymbols(1,12, custom_init=custom_init) sage: M.customize 'hi' sage: M = ModularSymbols(1,12, custom_init=custom_init, use_cache=False) sage: M.customize 'hi2'
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sage.modular.modsym.modsym.ModularSymbols_clear_cache()¶ Clear the global cache of modular symbols spaces.
EXAMPLES:
sage: sage.modular.modsym.modsym.ModularSymbols_clear_cache() sage: sorted(sage.modular.modsym.modsym._cache) [] sage: M = ModularSymbols(6,2) sage: sorted(sage.modular.modsym.modsym._cache) [(Congruence Subgroup Gamma0(6), 2, 0, Rational Field)] sage: sage.modular.modsym.modsym.ModularSymbols_clear_cache() sage: sorted(sage.modular.modsym.modsym._cache) []
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sage.modular.modsym.modsym.canonical_parameters(group, weight, sign, base_ring)¶ Return the canonically normalized parameters associated to a choice of group, weight, sign, and base_ring. That is, normalize each of these to be of the correct type, perform all appropriate type checking, etc.
EXAMPLES:
sage: p1 = sage.modular.modsym.modsym.canonical_parameters(5,int(2),1,QQ) ; p1 (Congruence Subgroup Gamma0(5), 2, 1, Rational Field) sage: p2 = sage.modular.modsym.modsym.canonical_parameters(Gamma0(5),2,1,QQ) ; p2 (Congruence Subgroup Gamma0(5), 2, 1, Rational Field) sage: p1 == p2 True sage: type(p1[1]) <type 'sage.rings.integer.Integer'>