Module: sage.modular.modsym.modsym
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
Module-level Functions
[group=True], [weight=Rational Field], [sign=0], [base_ring=2], [use_cache=1]) |
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. This is ignored if group is a Dirichlet character.
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
:
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
.
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 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(); f x^4 + (-zeta6 - 1)*x^3 + (-8*zeta6)*x^2 + (10*zeta6 - 5)*x + 21*zeta6 - 21 sage: f.factor() (x + -2*zeta6 - 1) * (x + -zeta6 - 2) * (x + zeta6 + 1)^2
More examples of spaces with character:
sage: e = DirichletGroup(5, RationalField()).gen(); e [-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^2 - 1 sage: m = ModularSymbols(e, 6); m.dimension() 6 sage: m.T(2).charpoly() x^6 - 873*x^4 - 82632*x^2 - 1860496
) |
group, weight, sign, base_ring) |
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