The module modular.modsym.modsym defines the following methods:
[group=Rational Field], [weight=0], [sign=2], [base_field=1]) |
Create an ambient space of modular symbols.
INPUT: group -- The 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_field - field (default is the field of rational numbers). The base field is ignored if group is a 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) Full Modular Symbols space for Gamma_0(11) of weight 4 and dimension 4 over Rational Field
We create some spaces for
.
sage: ModularSymbols(Gamma1(13),2) Full Modular Symbols space for Gamma_1(13) of weight 2 and dimension 15 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).gen()**2 sage: e.order() 6 sage: M = ModularSymbols(e, 2); M Full Modular Symbols space of level 13, weight 2, character [zeta_12^2] and dimension 4 over Cyclotomic Field of order 12 and degree 4 sage: f = M.T(2).charpoly(); f x^4 + (-zeta_12^2 - 1)*x^3 + (-8*zeta_12^2)*x^2 + (10*zeta_12^2 - 5)*x + 21*zeta_12^2 - 21 sage: f.factor() [(x + -2*zeta_12^2 - 1, 1), (x + -zeta_12^2 - 2, 1), (x + zeta_12^2 + 1, 2)]
More examples of spaces with character:
sage: e = DirichletGroup(5, RationalField()).gen(); e [-1] sage: m = ModularSymbols(e, 2); m Full Modular Symbols space of level 5, weight 2, character [-1] and dimension 2 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
See About this document... for information on suggesting changes.