27.7 Space of boundary modular symbols

Module: sage.modular.modsym.boundary

Used mainly for computing the cuspidal subspace of modular symbols.

Module-level Functions

is_FreeModuleElement( x)

Class: BoundarySpace

class BoundarySpace
Space of boundary symbols for a congruence subgroup of SL_2(Z).

This class is an abstract base class, so only derived classes should be instantiated.

INPUT: weight - int, the weight group - congroup.CongruenceGroup, a congruence subgroup. sign - int, either -1, 0, or 1 base_ring - rings.Ring (defaults to the rational numbers)

BoundarySpace( self, [group=None], [weight=Rational Field], [sign=0], [base_ring=2], [character=Congruence Subgroup Gamma0(1)])

Initialize a space of boundary symbols.

Functions: character,$  $ element,$  $ filename,$  $ free_module,$  $ gen,$  $ group,$  $ is_ambient,$  $ rank,$  $ sign,$  $ weight

character( self)

Return the Dirichlet character assocaited to this space of boundary modular symbols.

element( self, x)

Creates and returns an element of self from a modular or sage symbol, if possible.

INPUT:
   x -- an object of one of the following types:
        ModularSymbol, ManinSymbol.  We ignore the group,
        weight, and character of the input modular or sage symbol.
OUTPUT:
   ModularSymbol -- a modular symbol with parent self.

filename( self)

Returns the filename of self that should be used to store self in the database.

INPUT:
   self -- space of modular symbols
OUTPUT:
   str -- a string

gen( self, [i=0])

Return the i-th generator of this space.

group( self)

Return the congruence subgroup associated to this space of boundary modular symbols.

rank( self)

The rank of the space generated by boundary symbols that have been found so far in the course of computing the boundary map.

WARNING: This number may change as more elements are coerced into this space!! (This is an implementation detail that will likely change.)

sign( self)

Return the sign of the complex conjugation involution on this space of boundary modular symbols.

weight( self)

Return the weight of this space of boundary modular symbols.

Special Functions: __call__,$  $ __len__,$  $ _coerce_in_manin_symbol,$  $ _cusp_index,$  $ _repr_

__call__( self, x)

Coerce x into a boundary symbol space.

If x is a modular symbol (with the same group, weight, character, sign, and base field), this returns the image of that modular symbol under the boundary map.

Class: BoundarySpace_wtk_eps

class BoundarySpace_wtk_eps
BoundarySpace_wtk_eps( self, eps, weight, [sign=0])

Space of boundary 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

Special Functions: _coerce_cusp,$  $ _cusp_index,$  $ _is_equiv,$  $ _repr_

_coerce_cusp( self, c)

Coerce symbol into a boundary symbol space.

Class: BoundarySpace_wtk_g0_Q

class BoundarySpace_wtk_g0_Q
Boundary symbols for Gamma_0(N) of integer weight k > 2 over the field Q.
BoundarySpace_wtk_g0_Q( self, level, weight, [sign=0])

Initialize a space of boundary symbols of weight k for Gamma_0(N), over Q.

For weight 2, it is faster to use BoundarySpace_wt2_g0_Q.

INPUT:
    level -- int, the level
    weight -- integer weight >= 2.
    sign -- int, either -1, 0, or 1

Special Functions: _coerce_cusp,$  $ _is_equiv,$  $ _repr_

_coerce_cusp( self, c)

Coerce cusp into a boundary symbol space.

Class: BoundarySpace_wtk_g1_Q

class BoundarySpace_wtk_g1_Q
BoundarySpace_wtk_g1_Q( self, level, weight, [sign=0])

Initialize a space of boundayr modular symbols for Gamma1(N).

INPUT:
    level -- int, the level
    weight -- int, the weight >= 2
    sign -- int, either -1, 0, or 1

sage: from sage.modular.modsym.boundary import BoundarySpace_wtk_g1_Q
sage: BoundarySpace_wtk_g1_Q(17,2)
Boundary Modular Symbols space for Gamma_1(17) of weight 2 over Rational
Field

Special Functions: _coerce_cusp,$  $ _cusp_index,$  $ _is_equiv,$  $ _repr_

_coerce_cusp( self, c)

Coerce symbol into a boundary symbol space.

Class: BoundarySpaceElement

class BoundarySpaceElement
A boundary symbol.
BoundarySpaceElement( self, parent, x)

Create a boundary symbol.

INPUT:
    parent -- BoundarySpace; a space of boundary modular symbols
    x -- a dict with integer keys and values in the base
         field of parent.

Functions: element,$  $ parent

Special Functions: __add__,$  $ __mul__,$  $ __neg__,$  $ __sub__,$  $ _repr_

_repr_( self)

Returns a string representation for self for printing purposes.

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