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.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 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 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
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 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 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 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(); 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'
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)
[]
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)
<type 'sage.rings.integer.Integer'>