Modular symbols {alpha, beta}

The ModularSymbol class represents a single modular symbol \(X^i Y^{k-2-i} \{\alpha, \beta\}\).

AUTHOR:

  • William Stein (2005, 2009)

TESTS:

sage: s = ModularSymbols(11).2.modular_symbol_rep()[0][1]; s
{-1/9, 0}
sage: loads(dumps(s)) == s
True
class sage.modular.modsym.modular_symbols.ModularSymbol(space, i, alpha, beta)

Bases: sage.structure.sage_object.SageObject

The modular symbol \(X^i\cdot Y^{k-2-i}\cdot \{\alpha, \beta\}\).

alpha()

For a symbol of the form \(X^i Y^{k-2-i}\{\alpha, \beta\}\), return \(\alpha\).

EXAMPLES:

sage: s = ModularSymbols(11,4).1.modular_symbol_rep()[0][1]; s
X^2*{-1/6, 0}
sage: s.alpha()
-1/6
sage: type(s.alpha())
<class 'sage.modular.cusps.Cusp'>
apply(g)

Act on this symbol by the element \(g \in {\rm GL}_2(\QQ)\).

INPUT:

  • g – a list [a,b,c,d], corresponding to the 2x2 matrix \(\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in {\rm GL}_2(\QQ)\).

OUTPUT:

  • FormalSum – a formal sum \(\sum_i c_i x_i\), where \(c_i\) are scalars and \(x_i\) are ModularSymbol objects, such that the sum \(\sum_i c_i x_i\) is the image of this symbol under the action of g. No reduction is performed modulo the relations that hold in self.space().

The action of \(g\) on symbols is by

\[P(X,Y)\{\alpha, \beta\} \mapsto P(dX-bY, -cx+aY) \{g(\alpha), g(\beta)\}.\]

Note that for us we have \(P=X^i Y^{k-2-i}\), which simplifies computation of the polynomial part slightly.

EXAMPLES:

sage: s = ModularSymbols(11,2).1.modular_symbol_rep()[0][1]; s
{-1/8, 0}
sage: a=1;b=2;c=3;d=4; s.apply([a,b,c,d])
{15/29, 1/2}
sage: x = -1/8;  (a*x+b)/(c*x+d)
15/29
sage: x = 0;  (a*x+b)/(c*x+d)
1/2
sage: s = ModularSymbols(11,4).1.modular_symbol_rep()[0][1]; s
X^2*{-1/6, 0}
sage: s.apply([a,b,c,d])
16*X^2*{11/21, 1/2} - 16*X*Y*{11/21, 1/2} + 4*Y^2*{11/21, 1/2}
sage: P = s.polynomial_part()
sage: X,Y = P.parent().gens()
sage: P(d*X-b*Y, -c*X+a*Y)
16*X^2 - 16*X*Y + 4*Y^2
sage: x=-1/6; (a*x+b)/(c*x+d)
11/21
sage: x=0; (a*x+b)/(c*x+d)
1/2
sage: type(s.apply([a,b,c,d]))
<class 'sage.structure.formal_sum.FormalSum'>
beta()

For a symbol of the form \(X^i Y^{k-2-i}\{\alpha, \beta\}\), return \(\beta\).

EXAMPLES:

sage: s = ModularSymbols(11,4).1.modular_symbol_rep()[0][1]; s
X^2*{-1/6, 0}
sage: s.beta()
0
sage: type(s.beta())
<class 'sage.modular.cusps.Cusp'>
i()

For a symbol of the form \(X^i Y^{k-2-i}\{\alpha, \beta\}\), return \(i\).

EXAMPLES:

sage: s = ModularSymbols(11).2.modular_symbol_rep()[0][1]
sage: s.i()
0
sage: s = ModularSymbols(1,28).0.modular_symbol_rep()[0][1]; s
X^22*Y^4*{0, Infinity}
sage: s.i()
22
manin_symbol_rep()

Returns a representation of self as a formal sum of Manin symbols. (The result is not cached.)

EXAMPLES:

sage: M = ModularSymbols(11,4)
sage: s = M.1.modular_symbol_rep()[0][1]; s
X^2*{-1/6, 0}
sage: s.manin_symbol_rep()
-[Y^2,(1,1)] - 2*[X*Y,(-1,0)] - [X^2,(-6,1)] - [X^2,(-1,0)]
sage: M(s.manin_symbol_rep()) == M([2,-1/6,0])
True
polynomial_part()

Return the polynomial part of this symbol, i.e. for a symbol of the form \(X^i Y^{k-2-i}\{\alpha, \beta\}\), return \(X^i Y^{k-2-i}\).

EXAMPLES:

sage: s = ModularSymbols(11).2.modular_symbol_rep()[0][1]
sage: s.polynomial_part()
1
sage: s = ModularSymbols(1,28).0.modular_symbol_rep()[0][1]; s
X^22*Y^4*{0, Infinity}
sage: s.polynomial_part()
X^22*Y^4
space()

The list of Manin symbols to which this symbol belongs.

EXAMPLES:

sage: s = ModularSymbols(11).2.modular_symbol_rep()[0][1]
sage: s.space()
Manin Symbol List of weight 2 for Gamma0(11)
weight()

Return the weight of the modular symbols space to which this symbol belongs; i.e. for a symbol of the form \(X^i Y^{k-2-i}\{\alpha, \beta\}\), return \(k\).

EXAMPLES:

sage: s = ModularSymbols(1,28).0.modular_symbol_rep()[0][1]
sage: s.weight()
28

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