12.2.2.3 ModularSymbolsAmbient_wtk_eps Objects

class ModularSymbolsAmbient_wtk_eps
ModularSymbolsAmbient_wtk_eps( eps, weight, [sign=0])

Space of modular symbols with given weight, character, and sign.

INPUT:
    eps -- dirichlet.DirichletCharacter, the "Nebentypus" character.
    weight -- int, the weight >= 2
    sign -- int, either -1, 0, or 1

sage: eps = DirichletGroup(4).gen(0)
sage: eps.order()
2
sage: ModularSymbols(eps, 2)
Full Modular Symbols space of level 4, weight 2, character [-1], sign 0,
and dimension 0 over Rational Field
sage: ModularSymbols(eps, 3)
Full Modular Symbols space of level 4, weight 3, character [-1], sign 0,
and dimension 2 over Rational Field

We next create a space with character of order bigger than 2.

sage: eps = DirichletGroup(5).gen(0)
sage: eps     # has order 4
[zeta_4]
sage: ModularSymbols(eps, 2).dimension()
0
sage: ModularSymbols(eps, 3).dimension()
2

Instances of class ModularSymbolsAmbient_wtk_eps have the following methods (in addition to inherited methods and special methods):

boundary_space,$  $ manin_symbols,$  $ modular_symbols_of_level,$  $ modular_symbols_of_sign,$  $ modular_symbols_of_weight

Further documentation:

modular_symbols_of_level( N)

Returns a space of modular symbols with the same defining properties (weight, sign, etc.) as this space except with the level N.

For example, if self is the space of modular symbols of weight 2 for Gamma_0(22), and level is 11, then this function returns modular symbols of weight 2 for Gamma_0(11).

modular_symbols_of_sign( sign)

Returns a space of modular symbols with the same defining properties (weight, sign, etc.) as this space except with given sign.

modular_symbols_of_weight( k)

Returns a space of modular symbols with the same defining properties (weight, sign, etc.) as this space except with weight k.

Instances of class ModularSymbolsAmbient_wtk_eps also have the following special methods:

__repr__,$  $ _cuspidal_new_subspace_dimension_formula,$  $ _cuspidal_submodule_dimension_formula,$  $ _degeneracy_raising_matrix,$  $ _dimension_formula,$  $ _matrix_of_operator_on_modular_symbols

Further documentation:

_matrix_of_operator_on_modular_symbols( codomain, R, [character_twist=False])

INPUT:
    self -- this space of modular symbols
    codomain -- space of modular symbols
    R -- list of lists [a,b,c,d] of length 4, which we view as elements of
GL_2(Q).
    
OUTPUT:
    a matrix, which represents the operator
    $$
       x \mapsto \sum_{g in R} g.x
    $$
    where g.x is the formal linear fractional transformation on modular
symbols.

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