The module modular.modsym.modsym defines the following functions:
[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 with sign 1 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 with sign 0 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_6], sign 0, and dimension 4 over Cyclotomic Field of order 6 and degree 2 sage: f = M.T(2).charpoly(); f x^4 + (-zeta_6 - 1)*x^3 + (-8*zeta_6)*x^2 + (10*zeta_6 - 5)*x + 21*zeta_6 - 21 sage: f.factor() [(x + -2*zeta_6 - 1, 1), (x + -zeta_6 - 2, 1), (x + zeta_6 + 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], sign 0, 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
[mode=manin]) |
Set the mode for printing of elements of modular symbols spaces.
INPUT: mode -- a string. The possibilities are as follows: * 'manin' -- (the default) formal sums of Manin symbols [P(X,Y),(u,v)] * 'modular' -- formal sums of Modular symbols P(X,Y)*{alpha,beta}, where alpha and beta are cusps * 'vector' -- as vectors on the basis for the ambient space
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