def xi(self, A):
r""" The eight-root of unity in front of the Weil representation.
INPUT:
-''N'' -- integer
-''A'' -- element of PSL(2,Z)
EXAMPLES::
sage: A=SL2Z([41,77,33,62])
sage: WR.xi(A)
-zeta8^3]
sage: S,T=SL2Z.gens()
sage: WR.xi(S)
-zeta8^3
sage: WR.xi(T)
1
sage: A=SL2Z([-1,1,-4,3])
sage: WR.xi(A)
-zeta8^2
sage: A=SL2Z([0,1,-1,0])
sage: WR.xi(A)
-zeta8
"""
a = Integer(A[0, 0])
b = Integer(A[0, 1])
c = Integer(A[1, 0])
d = Integer(A[1, 1])
if (c == 0):
return 1
z = CyclotomicField(8).gen()
N = self._N
N2 = odd_part(N)
Neven = ZZ(2 * N).divide_knowing_divisible_by(N2)
c2 = odd_part(c)
Nc = gcd(Integer(2 * N), Integer(c))
cNc = ZZ(c).divide_knowing_divisible_by(Nc)
f1 = kronecker(-a, cNc)
f2 = kronecker(cNc, ZZ(2 * N).divide_knowing_divisible_by(Nc))
if (is_odd(c)):
s = c * N2
elif (c % Neven == 0):
s = (c2 + 1 - N2) * (a + 1)
else:
s = (c2 + 1 - N2) * (a + 1) - N2 * a * c2
r = -1 - QQ(N2) / QQ(gcd(c, N2)) + s
xi = f1 * f2 * z**r
return xi
def xi(self,A):
r""" The eight-root of unity in front of the Weil representation.
INPUT:
-''N'' -- integer
-''A'' -- element of PSL(2,Z)
EXAMPLES::
sage: A=SL2Z([41,77,33,62])
sage: WR.xi(A)
-zeta8^3]
sage: S,T=SL2Z.gens()
sage: WR.xi(S)
-zeta8^3
sage: WR.xi(T)
1
sage: A=SL2Z([-1,1,-4,3])
sage: WR.xi(A)
-zeta8^2
sage: A=SL2Z([0,1,-1,0])
sage: WR.xi(A)
-zeta8
"""
a=Integer(A[0,0]); b=Integer(A[0,1])
c=Integer(A[1,0]); d=Integer(A[1,1])
if(c==0):
return 1
z=CyclotomicField(8).gen()
N=self._N
N2=odd_part(N)
Neven=ZZ(2*N).divide_knowing_divisible_by(N2)
c2=odd_part(c)
Nc=gcd(Integer(2*N),Integer(c))
cNc=ZZ(c).divide_knowing_divisible_by(Nc)
f1=kronecker(-a,cNc)
f2=kronecker(cNc,ZZ(2*N).divide_knowing_divisible_by(Nc))
if(is_odd(c)):
s=c*N2
elif( c % Neven == 0):
s=(c2+1-N2)*(a+1)
else:
s=(c2+1-N2)*(a+1)-N2*a*c2
r=-1-QQ(N2)/QQ(gcd(c,N2))+s
xi=f1*f2*z**r
return xi
def _dimension_formula(self,k,eps=1,cuspidal=1):
ep = 0
N = self._N
if (2*k) % 4 == 1: ep = 1
if (2*k) % 4 == 3: ep = -1
if ep==0: return 0,0
if eps==-1:
ep = -ep
twok = ZZ(2*k)
K0 = 1
sqf = ZZ(N).divide_knowing_divisible_by(squarefree_part(N))
if sqf>12:
b2 = max(sqf.divisors())
else:
b2 = 1
b = sqrt(b2)
if ep==1:
K0 = floor(QQ(b+2)/QQ(2))
else:
# print "b=",b
K0 = floor(QQ(b-1)/QQ(2))
if is_even(N):
e2 = ep*kronecker(2,twok)/QQ(4)
else:
e2 = 0
N2 = odd_part(N)
N22 = ZZ(N).divide_knowing_divisible_by(N2)
k3 = kronecker(3,twok)
if gcd(3,N)>1:
if eps==1:
e3 = -ep*kronecker(-3,4*k+ep-1)/QQ(3)
else:
e3 = -1*ep*kronecker(-3,4*k+ep+1)/QQ(3)
#e3 = -1/3*ep
else:
f1 = kronecker(3,2*N22)*kronecker(-12,N2) - ep
f2 = kronecker(-3,twok+1)
e3 = f1*f2/QQ(6)
ID = QQ(N+ep)*(k-1)/QQ(12)
P = 0
for d in ZZ(4*N).divisors():
dm4=d % 4
if dm4== 2 or dm4 == 1:
h = 0
elif d == 3:
h = QQ(1)/QQ(3)
elif d == 4:
h = QQ(1)/QQ(2)
else:
h = class_nr_pos_def_qf(-d)
if self._verbose>1:
print "h({0})={1}".format(d,h)
if h<>0:
P= P + h
P = QQ(P)/QQ(4)
if self._verbose>0:
print "P=",P
P=P + QQ(ep)*kronecker(-4,N)/QQ(8)
if eps==-1:
P = -P
if self._verbose>0:
print "P=",P
# P = -2*N**2 + N*(twok+10-ep*3) +(twok+10)*ep-1
if self._verbose>0:
print "ID=",ID
P = P - QQ(1)/QQ(2*K0)
# P = QQ(P)/QQ(24) - K0
# P = P - K0
res = ID + P + e2 + e3
if self._verbose>1:
print "twok=",twok
print "K0=",K0
print "ep=",ep
print "e2=",e2
print "e3=",e3
print "P=",P
if cuspidal==0:
res = res + K0
return res #,ep
def _dimension_formula(self,k,eps=1,cuspidal=1):
ep = 0
N = self._N
if (2*k) % 4 == 1: ep = 1
if (2*k) % 4 == 3: ep = -1
if ep==0: return 0,0
if eps==-1:
ep = -ep
twok = ZZ(2*k)
K0 = 1
sqf = ZZ(N).divide_knowing_divisible_by(squarefree_part(N))
if sqf>12:
b2 = max(sqf.divisors())
else:
b2 = 1
b = sqrt(b2)
if ep==1:
K0 = floor(QQ(b+2)/QQ(2))
else:
# print "b=",b
K0 = floor(QQ(b-1)/QQ(2))
if is_even(N):
e2 = ep*kronecker(2,twok)/QQ(4)
else:
e2 = 0
N2 = odd_part(N)
N22 = ZZ(N).divide_knowing_divisible_by(N2)
k3 = kronecker(3,twok)
if gcd(3,N)>1:
if eps==1:
e3 = -ep*kronecker(-3,4*k+ep-1)/QQ(3)
else:
e3 = -1*ep*kronecker(-3,4*k+ep+1)/QQ(3)
#e3 = -1/3*ep
else:
f1 = kronecker(3,2*N22)*kronecker(-12,N2) - ep
f2 = kronecker(-3,twok+1)
e3 = f1*f2/QQ(6)
ID = QQ(N+ep)*(k-1)/QQ(12)
P = 0
for d in ZZ(4*N).divisors():
dm4=d % 4
if dm4== 2 or dm4 == 1:
h = 0
elif d == 3:
h = QQ(1)/QQ(3)
elif d == 4:
h = QQ(1)/QQ(2)
else:
h = class_nr_pos_def_qf(-d)
if self._verbose>1:
print("h({0})={1}".format(d,h))
if h!=0:
P= P + h
P = QQ(P)/QQ(4)
if self._verbose>0:
print("P={0}".format(P))
P=P + QQ(ep)*kronecker(-4,N)/QQ(8)
if eps==-1:
P = -P
if self._verbose>0:
print("P={0}".format(P))
# P = -2*N**2 + N*(twok+10-ep*3) +(twok+10)*ep-1
if self._verbose>0:
print("ID={0}".format(ID))
P = P - QQ(1)/QQ(2*K0)
# P = QQ(P)/QQ(24) - K0
# P = P - K0
res = ID + P + e2 + e3
if self._verbose>1:
print("twok={0}".format(twok))
print("K0={0}".format(K0))
print("ep={0}".format(ep))
print("e2={0}".format(e2))
print("e3={0}".format(e3))
print("P={0}".format(P))
if cuspidal==0:
res = res + K0
return res #,ep