def __repr__(self):
s="Multiplier system defined by action on the generators:"
if self._group==SL2Z:
S,T=SL2Z.gens()
Z=S*S
s+="r(S)=",self.vals[S]
s+="r(T)=",self.vals[T]
s+="r(Z)=",self.vals[Z]
else:
for g in self.vals.keys():
s+="r(",g,")=",self.vals[g]
return s
def __repr__(self):
s = "Multiplier system defined by action on the generators:"
if self._group == SL2Z:
S, T = SL2Z.gens()
Z = S * S
s += "r(S)=", self.vals[S]
s += "r(T)=", self.vals[T]
s += "r(Z)=", self.vals[Z]
else:
for g in self.vals.keys():
s += "r(", g, ")=", self.vals[g]
return s
def get_rk(self, n, k):
r"""
Compute the coefficient r_k((f|A_n))
"""
if not self._rks.has_key((n, k)):
S, T = SL2Z.gens()
nstar = self._get_shifted_coset_m(n, S)
kstar = self._k - 2 - k
i1 = self.get_integral_from_1_to_oo(n, k)
i2 = self.get_integral_from_1_to_oo(nstar, kstar)
if self._verbose > 0:
print "i1=", i1
print "i2=", i2
self._rks[(n, k)] = i1 + i2 * (-1)**(k + 1)
return self._rks[(n, k)]
def get_rk(self, n, k):
r"""
Compute the coefficient r_k((f|A_n))
"""
if (n, k) not in self._rks:
S, T = SL2Z.gens()
nstar = self._get_shifted_coset_m(n, S)
kstar = self._k - 2 - k
i1 = self.get_integral_from_1_to_oo(n, k)
i2 = self.get_integral_from_1_to_oo(nstar, kstar)
if self._verbose > 0:
print("i1={0}".format(i1))
print("i2={0}".format(i2))
self._rks[(n, k)] = i1 + i2 * (-1)**(k + 1)
return self._rks[(n, k)]
def get_rk(self,n,k):
r"""
Compute the coefficient r_k((f|A_n))
"""
if not self._rks.has_key((n,k)):
S,T = SL2Z.gens()
nstar = self._get_shifted_coset_m(n,S)
kstar = self._k-2-k
i1 = self.get_integral_from_1_to_oo(n,k)
i2 = self.get_integral_from_1_to_oo(nstar,kstar)
if self._verbose>0:
print "i1=",i1
print "i2=",i2
self._rks[(n,k)] = i1+i2*(-1)**(k+1)
return self._rks[(n,k)]
def __init__(self,G,v=None,**kwargs):
r"""
G should be a subgroup of PSL(2,Z).
EXAMPLE::
sage: te=TestMultiplier(Gamma0(2),weight=1/2)
sage: r=InducedRepresentation(Gamma0(2),v=te)
"""
dim = len(list(G.coset_reps()))
MultiplierSystem.__init__(self,Gamma0(1),dimension=dim)
self._induced_from=G
# setup the action on S and T (should be faster...)
self.v = v
if v<>None:
k = v.order()
if k>2:
K = CyclotomicField(k)
else:
K=ZZ
self.S=matrix(K,dim,dim)
self.T=matrix(K,dim,dim)
else:
self.S=matrix(dim,dim)
self.T=matrix(dim,dim)
S,T=SL2Z.gens()
if hasattr(G,"coset_reps"):
if isinstance(G.coset_reps(),list):
Vl=G.coset_reps()
else:
Vl=list(G.coset_reps())
elif hasattr(G,"_G"):
Vl=list(G._G.coset_reps())
else:
raise ValueError,"Could not get coset representatives from {0}!".format(G)
self.repsT=dict()
self.repsS=dict()
for i in range(dim):
Vi=Vl[i]
for j in range(dim):
Vj=Vl[j]
BS = Vi*S*Vj**-1
BT = Vi*T*Vj**-1
#print "i,j
#print "ViSVj^-1=",BS
#print "ViTVj^-1=",BT
if BS in G:
if v<>None:
vS=v(BS)
else:
vS=1
self.S[i,j]=vS
self.repsS[(i,j)]=BS
if BT in G:
if v<>None:
vT=v(BT)
else:
vT=1
self.T[i,j]=vT
self.repsT[(i,j)]=BT
def nice_coset_reps(G):
r"""
Compute a better/nicer list of right coset representatives [V_j]
i.e. SL2Z = \cup G V_j
Use this routine for known congruence subgroups.
EXAMPLES::
sage: G=MySubgroup(Gamma0(5))
sage: G._get_coset_reps_from_G(Gamma0(5))
[[1 0]
[0 1], [ 0 -1]
[ 1 0], [ 0 -1]
[ 1 1], [ 0 -1]
[ 1 -1], [ 0 -1]
[ 1 2], [ 0 -1]
[ 1 -2]]
"""
cl=list()
S,T=SL2Z.gens()
lvl=G.generalised_level()
# Start with identity rep.
cl.append(SL2Z([1 ,0 ,0 ,1 ]))
if(not S in G):
cl.append(S)
# If the original group is given as a Gamma0 then
# the reps are not the one we want
# I.e. we like to have a fundamental domain in
# -1/2 <=x <= 1/2 for Gamma0, Gamma1, Gamma
for j in range(1 , ZZ( ceil(RR(lvl/2.0))+2)):
for ep in [1 ,-1 ]:
if(len(cl)>=G.index()):
break
# The ones about 0 are all of this form
A=SL2Z([0 ,-1 ,1 ,ep*j])
# just make sure they are inequivalent
try:
for V in cl:
if((A<>V and A*V**-1 in G) or cl.count(A)>0 ):
raise StopIteration()
cl.append(A)
except StopIteration:
pass
# We now addd the rest of the "flips" of these reps.
# So that we end up with a connected domain
i=1
while(True):
lold=len(cl)
for V in cl:
for A in [S,T,T**-1 ]:
B=V*A
try:
for W in cl:
if( (B*W**-1 in G) or cl.count(B)>0 ):
raise StopIteration()
cl.append(B)
except StopIteration:
pass
if(len(cl)>=G.index() or lold>=len(cl)):
# If we either did not addd anything or if we addded enough
# we exit
break
# If we missed something (which is unlikely)
if(len(cl)<>G.index()):
print "cl=",cl
raise ValueError,"Problem getting coset reps! Need %s and got %s" %(G.index(),len(cl))
return cl
def nice_coset_reps(G):
r"""
Compute a better/nicer list of right coset representatives [V_j]
i.e. SL2Z = \cup G V_j
Use this routine for known congruence subgroups.
EXAMPLES::
sage: G=MySubgroup(Gamma0(5))
sage: G._get_coset_reps_from_G(Gamma0(5))
[[1 0]
[0 1], [ 0 -1]
[ 1 0], [ 0 -1]
[ 1 1], [ 0 -1]
[ 1 -1], [ 0 -1]
[ 1 2], [ 0 -1]
[ 1 -2]]
"""
cl = list()
S, T = SL2Z.gens()
lvl = G.generalised_level()
# Start with identity rep.
cl.append(SL2Z([1, 0, 0, 1]))
if (not S in G):
cl.append(S)
# If the original group is given as a Gamma0 then
# the reps are not the one we want
# I.e. we like to have a fundamental domain in
# -1/2 <=x <= 1/2 for Gamma0, Gamma1, Gamma
for j in range(1, ZZ(ceil(RR(lvl / 2.0)) + 2)):
for ep in [1, -1]:
if (len(cl) >= G.index()):
break
# The ones about 0 are all of this form
A = SL2Z([0, -1, 1, ep * j])
# just make sure they are inequivalent
try:
for V in cl:
if ((A <> V and A * V**-1 in G) or cl.count(A) > 0):
raise StopIteration()
cl.append(A)
except StopIteration:
pass
# We now addd the rest of the "flips" of these reps.
# So that we end up with a connected domain
i = 1
while (True):
lold = len(cl)
for V in cl:
for A in [S, T, T**-1]:
B = V * A
try:
for W in cl:
if ((B * W**-1 in G) or cl.count(B) > 0):
raise StopIteration()
cl.append(B)
except StopIteration:
pass
if (len(cl) >= G.index() or lold >= len(cl)):
# If we either did not addd anything or if we addded enough
# we exit
break
# If we missed something (which is unlikely)
if (len(cl) <> G.index()):
print "cl=", cl
raise ValueError, "Problem getting coset reps! Need %s and got %s" % (
G.index(), len(cl))
return cl
def __init__(self, G, v=None, **kwargs):
r"""
G should be a subgroup of PSL(2,Z).
EXAMPLE::
sage: te=TestMultiplier(Gamma0(2),weight=1/2)
sage: r=InducedRepresentation(Gamma0(2),v=te)
"""
dim = len(list(G.coset_reps()))
MultiplierSystem.__init__(self, Gamma0(1), dimension=dim)
self._induced_from = G
# setup the action on S and T (should be faster...)
self.v = v
if v != None:
k = v.order()
if k > 2:
K = CyclotomicField(k)
else:
K = ZZ
self.S = matrix(K, dim, dim)
self.T = matrix(K, dim, dim)
else:
self.S = matrix(dim, dim)
self.T = matrix(dim, dim)
S, T = SL2Z.gens()
if hasattr(G, "coset_reps"):
if isinstance(G.coset_reps(), list):
Vl = G.coset_reps()
else:
Vl = list(G.coset_reps())
elif hasattr(G, "_G"):
Vl = list(G._G.coset_reps())
else:
raise ValueError(
"Could not get coset representatives from {0}!".format(G))
self.repsT = dict()
self.repsS = dict()
for i in range(dim):
Vi = Vl[i]
for j in range(dim):
Vj = Vl[j]
BS = Vi * S * Vj**-1
BT = Vi * T * Vj**-1
#print "i,j
#print "ViSVj^-1=",BS
#print "ViTVj^-1=",BT
if BS in G:
if v != None:
vS = v(BS)
else:
vS = 1
self.S[i, j] = vS
self.repsS[(i, j)] = BS
if BT in G:
if v != None:
vT = v(BT)
else:
vT = 1
self.T[i, j] = vT
self.repsT[(i, j)] = BT
