A single element of an ambient space of modular symbols

class sage.modular.modsym.element.ModularSymbolsElement(parent, x, check=True)

Bases: HeckeModuleElement

An element of a space of modular symbols.

list()

Return a list of the coordinates of self in terms of a basis for the ambient space.

EXAMPLES:

Sage

sage: ModularSymbols(37, 2).0.list()
[1, 0, 0, 0, 0]

Python

>>> from sage.all import *
>>> ModularSymbols(Integer(37), Integer(2)).gen(0).list()
[1, 0, 0, 0, 0]

Sage Live

ModularSymbols(37, 2).0.list()

manin_symbol_rep()

Return a representation of self as a formal sum of Manin symbols.

EXAMPLES:

Sage

sage: x = ModularSymbols(37, 4).0
sage: x.manin_symbol_rep()
[X^2,(0,1)]

Python

>>> from sage.all import *
>>> x = ModularSymbols(Integer(37), Integer(4)).gen(0)
>>> x.manin_symbol_rep()
[X^2,(0,1)]

Sage Live

x = ModularSymbols(37, 4).0
x.manin_symbol_rep()

The result is cached:

Sage

sage: x.manin_symbol_rep() is x.manin_symbol_rep()
True

Python

>>> from sage.all import *
>>> x.manin_symbol_rep() is x.manin_symbol_rep()
True

Sage Live

x.manin_symbol_rep() is x.manin_symbol_rep()

modular_symbol_rep()

Return a representation of self as a formal sum of modular symbols.

EXAMPLES:

Sage

sage: x = ModularSymbols(37, 4).0
sage: x.modular_symbol_rep()
X^2*{0, Infinity}

Python

>>> from sage.all import *
>>> x = ModularSymbols(Integer(37), Integer(4)).gen(0)
>>> x.modular_symbol_rep()
X^2*{0, Infinity}

Sage Live

x = ModularSymbols(37, 4).0
x.modular_symbol_rep()

The result is cached:

Sage

sage: x.modular_symbol_rep() is x.modular_symbol_rep()
True

Python

>>> from sage.all import *
>>> x.modular_symbol_rep() is x.modular_symbol_rep()
True

Sage Live

x.modular_symbol_rep() is x.modular_symbol_rep()

sage.modular.modsym.element.is_ModularSymbolsElement(x)

Return True if x is an element of a modular symbols space.

EXAMPLES:

Sage

sage: sage.modular.modsym.element.is_ModularSymbolsElement(ModularSymbols(11, 2).0)
doctest:warning...
DeprecationWarning: The function is_ModularSymbolsElement is deprecated;
use 'isinstance(..., ModularSymbolsElement)' instead.
See https://github.com/sagemath/sage/issues/38184 for details.
True
sage: sage.modular.modsym.element.is_ModularSymbolsElement(13)
False

Python

>>> from sage.all import *
>>> sage.modular.modsym.element.is_ModularSymbolsElement(ModularSymbols(Integer(11), Integer(2)).gen(0))
doctest:warning...
DeprecationWarning: The function is_ModularSymbolsElement is deprecated;
use 'isinstance(..., ModularSymbolsElement)' instead.
See https://github.com/sagemath/sage/issues/38184 for details.
True
>>> sage.modular.modsym.element.is_ModularSymbolsElement(Integer(13))
False

Sage Live

sage.modular.modsym.element.is_ModularSymbolsElement(ModularSymbols(11, 2).0)
sage.modular.modsym.element.is_ModularSymbolsElement(13)

sage.modular.modsym.element.set_modsym_print_mode(mode='manin')

Set the mode for printing of elements of modular symbols spaces.

INPUT:

• mode – 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

OUTPUT: none

EXAMPLES:

Sage

sage: M = ModularSymbols(13, 8)
sage: x = M.0 + M.1 + M.14
sage: set_modsym_print_mode('manin'); x
[X^5*Y,(1,11)] + [X^5*Y,(1,12)] + [X^6,(1,11)]
sage: set_modsym_print_mode('modular'); x
1610510*X^6*{-1/11, 0} + 893101*X^5*Y*{-1/11, 0} + 206305*X^4*Y^2*{-1/11, 0} + 25410*X^3*Y^3*{-1/11, 0} + 1760*X^2*Y^4*{-1/11, 0} + 65*X*Y^5*{-1/11, 0} - 248832*X^6*{-1/12, 0} - 103680*X^5*Y*{-1/12, 0} - 17280*X^4*Y^2*{-1/12, 0} - 1440*X^3*Y^3*{-1/12, 0} - 60*X^2*Y^4*{-1/12, 0} - X*Y^5*{-1/12, 0} + Y^6*{-1/11, 0}
sage: set_modsym_print_mode('vector'); x
(1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0)
sage: set_modsym_print_mode()

Python

>>> from sage.all import *
>>> M = ModularSymbols(Integer(13), Integer(8))
>>> x = M.gen(0) + M.gen(1) + M.gen(14)
>>> set_modsym_print_mode('manin'); x
[X^5*Y,(1,11)] + [X^5*Y,(1,12)] + [X^6,(1,11)]
>>> set_modsym_print_mode('modular'); x
1610510*X^6*{-1/11, 0} + 893101*X^5*Y*{-1/11, 0} + 206305*X^4*Y^2*{-1/11, 0} + 25410*X^3*Y^3*{-1/11, 0} + 1760*X^2*Y^4*{-1/11, 0} + 65*X*Y^5*{-1/11, 0} - 248832*X^6*{-1/12, 0} - 103680*X^5*Y*{-1/12, 0} - 17280*X^4*Y^2*{-1/12, 0} - 1440*X^3*Y^3*{-1/12, 0} - 60*X^2*Y^4*{-1/12, 0} - X*Y^5*{-1/12, 0} + Y^6*{-1/11, 0}
>>> set_modsym_print_mode('vector'); x
(1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0)
>>> set_modsym_print_mode()

Sage Live

M = ModularSymbols(13, 8)
x = M.0 + M.1 + M.14
set_modsym_print_mode('manin'); x
set_modsym_print_mode('modular'); x
set_modsym_print_mode('vector'); x
set_modsym_print_mode()

Make live