def apply(self, g):
r"""
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
.. MATH::
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'>
"""
space = self.__space
i = self.__i
k = space.weight()
a,b,c,d = tuple(g)
coeffs = apply_to_monomial(i, k-2, d, -b, -c, a)
g_alpha = self.__alpha.apply(g)
g_beta = self.__beta.apply(g)
return formal_sum.FormalSum([(coeffs[j], ModularSymbol(space, j, g_alpha, g_beta)) \
for j in reversed(range(k-1)) if coeffs[j] != 0])
def __manin_symbol_rep(self, alpha):
"""
Return Manin symbol representation of X^i*Y^(k-2-i){0,alpha}.
EXAMPLES::
sage: s = ModularSymbols(11,2).1.modular_symbol_rep()[0][1]; s
{-1/8, 0}
sage: s.manin_symbol_rep() # indirect doctest
-(-8,1) - (1,1)
sage: M = ModularSymbols(11,2)
sage: s = M( (1,9) ); s
(1,9)
sage: t = s.modular_symbol_rep()[0][1].manin_symbol_rep(); t
-(-9,1) - (1,1)
sage: M(t)
(1,9)
"""
space = self.__space
i = self.__i
k = space.weight()
v = [(0, 1), (1, 0)]
if not alpha.is_infinity():
v += [(x.numerator(), x.denominator())
for x in arith.convergents(alpha._rational_())]
sign = 1
apply = sage.modular.modsym.manin_symbols.apply_to_monomial
mansym = sage.modular.modsym.manin_symbols.ManinSymbol
z = formal_sum.FormalSum(0)
for j in range(1, len(v)):
c = sign * v[j][1]
d = v[j - 1][1]
coeffs = apply(i, k - 2, sign * v[j][0], v[j - 1][0],
sign * v[j][1], v[j - 1][1])
w = [(coeffs[j], mansym(space, (j, c, d))) \
for j in range(k-1) if coeffs[j] != 0]
z += formal_sum.FormalSum(w)
sign *= -1
return z
def __manin_symbol_rep(self, alpha):
"""
Return Manin symbol representation of X^i*Y^(k-2-i){0,alpha}.
EXAMPLES::
sage: s = ModularSymbols(11,2).1.modular_symbol_rep()[0][1]; s
{-1/8, 0}
sage: s.manin_symbol_rep() # indirect doctest
-(1,1) - (-8,1)
sage: M = ModularSymbols(11,2)
sage: s = M( (1,9) ); s
(1,9)
sage: t = s.modular_symbol_rep()[0][1].manin_symbol_rep(); t
-(1,1) - (-9,1)
sage: M(t)
(1,9)
"""
space = self.__space
i = self.__i
k = space.weight()
v = [(0,1), (1,0)]
if not alpha.is_infinity():
cf = alpha._rational_().continued_fraction()
v.extend((cf.p(k),cf.q(k)) for k in range(len(cf)))
sign = 1
z = formal_sum.FormalSum(0)
for j in range(1,len(v)):
c = sign*v[j][1]
d = v[j-1][1]
coeffs = apply_to_monomial(i, k-2, sign*v[j][0], v[j-1][0],
sign*v[j][1], v[j-1][1])
w = [(coeffs[j], ManinSymbol(space, (j, c, d)))
for j in range(k-1) if coeffs[j] != 0]
z += formal_sum.FormalSum(w)
sign *= -1
return z