def __call__(self, g):
"""
Evaluate ``self`` on a group element ``g``.
OUTPUT:
An element in
:meth:`~sage.groups.abelian_gps.dual_abelian_group.DualAbelianGroup_class.base_ring`.
EXAMPLES::
sage: F = AbelianGroup(5, [2,3,5,7,8], names="abcde")
sage: a,b,c,d,e = F.gens()
sage: Fd = F.dual_group(names="ABCDE")
sage: A,B,C,D,E = Fd.gens()
sage: A*B^2*D^7
A*B^2
sage: A(a)
-1
sage: B(b)
zeta840^140 - 1
sage: CC(B(b)) # abs tol 1e-8
-0.499999999999995 + 0.866025403784447*I
sage: A(a*b)
-1
"""
F = self.parent().base_ring()
expsX = self.exponents()
expsg = g.exponents()
order = self.parent().gens_orders()
N = LCM(order)
if is_ComplexField(F):
from sage.symbolic.constants import pi
I = F.gen()
PI = F(pi)
ans = prod([
exp(2 * PI * I * expsX[i] * expsg[i] / order[i])
for i in range(len(expsX))
])
return ans
ans = F(1) ## assumes F is the cyclotomic field
zeta = F.gen()
for i in range(len(expsX)):
order_noti = N / order[i]
ans = ans * zeta**(expsX[i] * expsg[i] * order_noti)
return ans
def __call__(self, g):
"""
Evaluate ``self`` on a group element ``g``.
OUTPUT:
An element in
:meth:`~sage.groups.abelian_gps.dual_abelian_group.DualAbelianGroup_class.base_ring`.
EXAMPLES::
sage: F = AbelianGroup(5, [2,3,5,7,8], names="abcde")
sage: a,b,c,d,e = F.gens()
sage: Fd = F.dual_group(names="ABCDE")
sage: A,B,C,D,E = Fd.gens()
sage: A*B^2*D^7
A*B^2
sage: A(a)
-1
sage: B(b)
zeta840^140 - 1
sage: CC(B(b)) # abs tol 1e-8
-0.499999999999995 + 0.866025403784447*I
sage: A(a*b)
-1
"""
F = self.parent().base_ring()
expsX = self.exponents()
expsg = g.exponents()
order = self.parent().gens_orders()
N = LCM(order)
if is_ComplexField(F):
from sage.symbolic.constants import pi
I = F.gen()
PI = F(pi)
ans = prod([exp(2*PI*I*expsX[i]*expsg[i]/order[i]) for i in range(len(expsX))])
return ans
ans = F(1) ## assumes F is the cyclotomic field
zeta = F.gen()
for i in range(len(expsX)):
order_noti = N/order[i]
ans = ans*zeta**(expsX[i]*expsg[i]*order_noti)
return ans
def __call__(self, g):
"""
Computes the value of a character self on a group element
g (g must be an element of self.group())
EXAMPLES:
sage: F = AbelianGroup(5, [2,3,5,7,8], names="abcde")
sage: a,b,c,d,e = F.gens()
sage: Fd = DualAbelianGroup(F, names="ABCDE")
sage: A,B,C,D,E = Fd.gens()
sage: A*B^2*D^7
A*B^2
sage: A(a) ## random last few digits
-1.0000000000000000 + 0.00000000000000013834419720915037*I
sage: B(b) ## random last few digits
-0.49999999999999983 + 0.86602540378443871*I
sage: A(a*b) ## random last few digits
-1.0000000000000000 + 0.00000000000000013834419720915037*I
"""
F = self.parent().base_ring()
expsX = list(self.list())
expsg = list(g.list())
invs = self.parent().invariants()
N = LCM(invs)
if is_ComplexField(F):
from sage.symbolic.constants import pi
I = F.gen()
PI = F(pi)
ans = prod([
exp(2 * PI * I * expsX[i] * expsg[i] / invs[i])
for i in range(len(expsX))
])
return ans
ans = F(1) ## assumes F is the cyclotomic field
zeta = F.gen()
#print F,zeta
for i in range(len(expsX)):
inv_noti = N / invs[i]
ans = ans * zeta**(expsX[i] * expsg[i] * inv_noti)
return ans
def __call__(self,g):
"""
Computes the value of a character self on a group element
g (g must be an element of self.group())
EXAMPLES:
sage: F = AbelianGroup(5, [2,3,5,7,8], names="abcde")
sage: a,b,c,d,e = F.gens()
sage: Fd = DualAbelianGroup(F, names="ABCDE")
sage: A,B,C,D,E = Fd.gens()
sage: A*B^2*D^7
A*B^2
sage: A(a) ## random last few digits
-1.0000000000000000 + 0.00000000000000013834419720915037*I
sage: B(b) ## random last few digits
-0.49999999999999983 + 0.86602540378443871*I
sage: A(a*b) ## random last few digits
-1.0000000000000000 + 0.00000000000000013834419720915037*I
"""
F = self.parent().base_ring()
expsX = list(self.list())
expsg = list(g.list())
invs = self.parent().invariants()
N = LCM(invs)
if is_ComplexField(F):
from sage.symbolic.constants import pi
I = F.gen()
PI = F(pi)
ans = prod([exp(2*PI*I*expsX[i]*expsg[i]/invs[i]) for i in range(len(expsX))])
return ans
ans = F(1) ## assumes F is the cyclotomic field
zeta = F.gen()
#print F,zeta
for i in range(len(expsX)):
inv_noti = N/invs[i]
ans = ans*zeta**(expsX[i]*expsg[i]*inv_noti)
return ans