27.1 Creation of modular symbols spaces

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

ModularSymbols( [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 $ \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 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

ModularSymbols_clear_cache( )

canonical_parameters( group, weight, sign, base_ring)

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