12.4.1.1 BoundarySpace Objects

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

Initialize a space of boundary symbols.

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

character,$  $ element,$  $ filename,$  $ free_module,$  $ gen,$  $ group,$  $ rank,$  $ sign,$  $ weight

Further documentation:

character( )

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

element( 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( )

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( [i=0])

Return the i-th generator of this space.

group( )

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

rank( )

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( )

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

weight( )

Return the weight of this space of boundary modular symbols.

Instances of class BoundarySpace also have the following special methods:

__call__,$  $ __len__,$  $ __repr__,$  $ _cusp_index

Further documentation:

__call__( 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.

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