Package sage :: Package modular :: Package modsym :: Module modsym
[show private | hide private]
[frames | no frames]

Module sage.modular.modsym.modsym

Creating modular symbols spaces
Function Summary
  ModularSymbols(group, weight, sign, base_field)
Create an ambient space of modular symbols.

Function Details

ModularSymbols(group=1, weight=2, sign=0, base_field=Rational Field)

Create an ambient space of modular symbols.

INPUT:
    group -- the congruence subgroup; one can also put a Dirichlet
        character eps here for modular symbols for eps
    weight -- int, the weight, which must be >= 2.
    sign -- the sign of the involution eta induced by complex conjugation.
            The default is 0, which means "no sign", i.e., take the
            whole space.
    base_field -- a field (default is the rational numbers)
            This is ignored if group is a character.
EXAMPLES:
First we create some spaces with trivial character:
    >>> from sage.modular.congroup import *
    >>> ModularSymbols(Gamma0(11),2).dimension()
    3
    >>> ModularSymbols(Gamma0(1),12).dimension()
    3

If we give an integer N for the congruence subgroup, it defaults
to Gamma0(N), since this is such a common case:
    >>> ModularSymbols(1,12,-1).dimension()
    1
    >>> 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 Gamma1(N).
    >>> ModularSymbols(Gamma1(13),2)
    Full Modular Symbols space for Gamma_1(13) of weight 2 and dimension 15 over Rational Field
    >>> ModularSymbols(Gamma1(13),2, sign=1).dimension()
    13
    >>> ModularSymbols(Gamma1(13),2, sign=-1).dimension()
    2
    >>> [ModularSymbols(Gamma1(7),k).dimension() for k in [2,3,4,5]]
    [5, 8, 12, 16]
    >>> ModularSymbols(Gamma1(5),11).dimension()
    20
    
We create a space with character:
    >>> from sage.modular.dirichlet import DirichletGroup
    >>> e = DirichletGroup(13).gen()**2
    >>> e.order()
    6
    >>> 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
    >>> 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
    >>> 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:
    >>> from sage.modular.dirichlet import DirichletGroup
    >>> from sage.rings.rings import RationalField
    >>> e = DirichletGroup(5, RationalField()).gen(); e
    [-1]
    >>> m = ModularSymbols(e, 2); m
    Full Modular Symbols space of level 5, weight 2, character [-1] and dimension 2 over Rational Field
    
    >>> m.T(2).charpoly()
    x^2 - 1
    >>> m = ModularSymbols(e, 6); m.dimension()
    6
    >>> m.T(2).charpoly()
    x^6 - 873*x^4 - 82632*x^2 - 1860496

Generated by Epydoc 2.1 on Fri May 20 19:41:03 2005 http://epydoc.sf.net