# A single element of an ambient space of modular symbols¶

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

An element of a space of modular symbols.

TESTS:

sage: x = ModularSymbols(3, 12).cuspidal_submodule().gen(0)
sage: x == loads(dumps(x))
True

list()

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

EXAMPLE:

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

manin_symbol_rep()

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

EXAMPLE:

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


The result is cached:

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

modular_symbol_rep()

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

EXAMPLE:

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


The result is cached:

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

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

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

EXAMPLES:

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

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

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

INPUT:

• mode - a 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

EXAMPLE:

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} - 248832*X^6*{-1/12, 0} + 893101*X^5*Y*{-1/11, 0} - 103680*X^5*Y*{-1/12, 0} + 206305*X^4*Y^2*{-1/11, 0} - 17280*X^4*Y^2*{-1/12, 0} + 25410*X^3*Y^3*{-1/11, 0} - 1440*X^3*Y^3*{-1/12, 0} + 1760*X^2*Y^4*{-1/11, 0} - 60*X^2*Y^4*{-1/12, 0} + 65*X*Y^5*{-1/11, 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()


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Subspace of ambient spaces of modular symbols

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Modular symbols {alpha, beta}