def apply_TT(self, j):
"""
Apply the matrix `TT=[-1,-1,0,1]` to the `j`-th Manin symbol.
INPUT:
- ``j`` - (int) a symbol index
OUTPUT: see documentation for apply()
EXAMPLES::
sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList_gamma0
sage: m = ManinSymbolList_gamma0(5,8)
sage: m.apply_TT(4)
[(38, 1)]
sage: [m.apply_TT(i) for i in range(10)]
[[(37, 1)],
[(41, 1)],
[(39, 1)],
[(40, 1)],
[(38, 1)],
[(36, 1)],
[(31, -1), (37, 1)],
[(35, -1), (41, 1)],
[(33, -1), (39, 1)],
[(34, -1), (40, 1)]]
"""
k = self._weight
i, u, v = self._symbol_list[j]
u, v = self.__syms.normalize(-u - v, u)
if (k - 2 - i) % 2 == 0:
s = 1
else:
s = -1
z = []
a = Integer(i)
for j in range(i + 1):
m = self.index((k - 2 - i + j, u, v))
z.append((m, s * a.binomial(j)))
s *= -1
return z
def apply_T(self, j):
"""
Apply the matrix `T=[0,1,-1,-1]` to the `j`-th Manin symbol.
INPUT:
- ``j`` - (int) a symbol index
OUTPUT: see documentation for apply()
EXAMPLES::
sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList_gamma0
sage: m = ManinSymbolList_gamma0(5,8)
sage: m.apply_T(4)
[(3, 1), (9, -6), (15, 15), (21, -20), (27, 15), (33, -6), (39, 1)]
sage: [m.apply_T(i) for i in range(10)]
[[(5, 1), (11, -6), (17, 15), (23, -20), (29, 15), (35, -6), (41, 1)],
[(0, 1), (6, -6), (12, 15), (18, -20), (24, 15), (30, -6), (36, 1)],
[(4, 1), (10, -6), (16, 15), (22, -20), (28, 15), (34, -6), (40, 1)],
[(2, 1), (8, -6), (14, 15), (20, -20), (26, 15), (32, -6), (38, 1)],
[(3, 1), (9, -6), (15, 15), (21, -20), (27, 15), (33, -6), (39, 1)],
[(1, 1), (7, -6), (13, 15), (19, -20), (25, 15), (31, -6), (37, 1)],
[(5, 1), (11, -5), (17, 10), (23, -10), (29, 5), (35, -1)],
[(0, 1), (6, -5), (12, 10), (18, -10), (24, 5), (30, -1)],
[(4, 1), (10, -5), (16, 10), (22, -10), (28, 5), (34, -1)],
[(2, 1), (8, -5), (14, 10), (20, -10), (26, 5), (32, -1)]]
"""
k = self._weight
i, u, v = self._symbol_list[j]
u, v = self.__syms.normalize(v, -u - v)
if (k - 2) % 2 == 0:
s = 1
else:
s = -1
z = []
a = Integer(k - 2 - i)
for j in range(k - 2 - i + 1):
m = self.index((j, u, v))
z.append((m, s * a.binomial(j)))
s *= -1
return z
def apply_TT(self, j):
"""
Apply the matrix `TT=[-1,-1,0,1]` to the `j`-th Manin symbol.
INPUT:
- ``j`` - (int) a symbol index
OUTPUT: see documentation for apply()
EXAMPLE::
sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList_gamma0
sage: m = ManinSymbolList_gamma0(5,8)
sage: m.apply_TT(4)
[(38, 1)]
sage: [m.apply_TT(i) for i in xrange(10)]
[[(37, 1)],
[(41, 1)],
[(39, 1)],
[(40, 1)],
[(38, 1)],
[(36, 1)],
[(31, -1), (37, 1)],
[(35, -1), (41, 1)],
[(33, -1), (39, 1)],
[(34, -1), (40, 1)]]
"""
k = self._weight
i, u, v = self._symbol_list[j]
u, v = self.__syms.normalize(-u-v,u)
if (k-2-i) % 2 == 0:
s = 1
else:
s = -1
z = []
a = Integer(i)
for j in range(i+1):
m = self.index((k-2-i+j, u, v))
z.append((m, s * a.binomial(j)))
s *= -1
return z
def apply_T(self, j):
"""
Apply the matrix `T=[0,1,-1,-1]` to the `j`-th Manin symbol.
INPUT:
- ``j`` - (int) a symbol index
OUTPUT: see documentation for apply()
EXAMPLE::
sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList_gamma0
sage: m = ManinSymbolList_gamma0(5,8)
sage: m.apply_T(4)
[(3, 1), (9, -6), (15, 15), (21, -20), (27, 15), (33, -6), (39, 1)]
sage: [m.apply_T(i) for i in xrange(10)]
[[(5, 1), (11, -6), (17, 15), (23, -20), (29, 15), (35, -6), (41, 1)],
[(0, 1), (6, -6), (12, 15), (18, -20), (24, 15), (30, -6), (36, 1)],
[(4, 1), (10, -6), (16, 15), (22, -20), (28, 15), (34, -6), (40, 1)],
[(2, 1), (8, -6), (14, 15), (20, -20), (26, 15), (32, -6), (38, 1)],
[(3, 1), (9, -6), (15, 15), (21, -20), (27, 15), (33, -6), (39, 1)],
[(1, 1), (7, -6), (13, 15), (19, -20), (25, 15), (31, -6), (37, 1)],
[(5, 1), (11, -5), (17, 10), (23, -10), (29, 5), (35, -1)],
[(0, 1), (6, -5), (12, 10), (18, -10), (24, 5), (30, -1)],
[(4, 1), (10, -5), (16, 10), (22, -10), (28, 5), (34, -1)],
[(2, 1), (8, -5), (14, 10), (20, -10), (26, 5), (32, -1)]]
"""
k = self._weight
i, u, v = self._symbol_list[j]
u, v = self.__syms.normalize(v,-u-v)
if (k-2) % 2 == 0:
s = 1
else:
s = -1
z = []
a = Integer(k-2-i)
for j in range(k-2-i+1):
m = self.index((j, u, v))
z.append((m, s * a.binomial(j)))
s *= -1
return z
def apply_TT(self, j):
"""
Apply the matrix `TT=[-1,-1,0,1]` to the `j`-th Manin symbol.
INPUT:
- ``j`` - (integer) a symbol index
OUTPUT:
A list of pairs `(j, c_i)`, where each `c_i` is an
integer, `j` is an integer (the `j`-th Manin symbol), and the
sum `c_i*x_i` is the image of self under the right action
of the matrix `T^2`.
EXAMPLES::
sage: eps = DirichletGroup(4).gen(0)
sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList_character
sage: m = ManinSymbolList_character(eps,2); m
Manin Symbol List of weight 2 for Gamma1(4) with character [-1]
sage: m.apply_TT(4)
[(0, 1)]
sage: [m.apply_TT(i) for i in range(len(m))]
[[(1, -1)], [(4, -1)], [(5, 1)], [(2, 1)], [(0, 1)], [(3, 1)]]
"""
k = self._weight
i, u, v = self._symbol_list[j]
u, v, r = self.__P1.normalize_with_scalar(-u - v, u)
r = self.__character(r)
if (k - 2 - i) % 2 == 0:
s = r
else:
s = -r
z = []
a = Integer(i)
for j in range(i + 1):
m, r = self.index((k - 2 - i + j, u, v))
z.append((m, s * r * a.binomial(j)))
s *= -1
return z
def apply_TT(self, j):
"""
Apply the matrix `TT=[-1,-1,0,1]` to the `j`-th Manin symbol.
INPUT:
- ``j`` - (integer) a symbol index
OUTPUT:
A list of pairs `(j, c_i)`, where each `c_i` is an
integer, `j` is an integer (the `j`-th Manin symbol), and the
sum `c_i*x_i` is the image of self under the right action
of the matrix `T^2`.
EXAMPLE::
sage: eps = DirichletGroup(4).gen(0)
sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList_character
sage: m = ManinSymbolList_character(eps,2); m
Manin Symbol List of weight 2 for Gamma1(4) with character [-1]
sage: m.apply_TT(4)
[(0, 1)]
sage: [m.apply_TT(i) for i in xrange(len(m))]
[[(1, -1)], [(4, -1)], [(5, 1)], [(2, 1)], [(0, 1)], [(3, 1)]]
"""
k = self._weight
i, u, v = self._symbol_list[j]
u, v, r = self.__P1.normalize_with_scalar(-u-v,u)
r = self.__character(r)
if (k-2-i) % 2 == 0:
s = r
else:
s = -r
z = []
a = Integer(i)
for j in range(i+1):
m, r = self.index((k-2-i+j, u, v))
z.append((m, s * r * a.binomial(j)))
s *= -1
return z
