def new_decomposition(self):
"""
Return complete irreducible Hecke decomposition of "new"
subspace of self.
OUTPUT:
- sorted Sequence of subspaces of self
EXAMPLES::
sage: from sage.modular.hilbert.sqrt5_hmf import F, QuaternionicModule
sage: H = QuaternionicModule(3 * F.prime_above(31)); H
Quaternionic module of dimension 6, level 15*a-6 (of norm 279=3^2*31) over QQ(sqrt(5))
sage: H.new_decomposition()
[
Subspace of dimension 1 ...,
Subspace of dimension 1 ...,
Subspace of dimension 1 ...,
Subspace of dimension 1 ...
]
"""
V = self.degeneracy_matrix().kernel()
primes = PrimesCoprimeTo(self.level())
p = primes.next()
T = self.hecke_matrix(p)
D = T.decomposition_of_subspace(V)
while len([X for X in D if not X[1]]) > 0:
p = primes.next()
verbose('Norm(p) = %s'%p.norm())
T = self.hecke_matrix(p)
D2 = []
for X in D:
if X[1]:
D2.append(X)
else:
for Z in T.decomposition_of_subspace(X[0]):
D2.append(Z)
D = D2
D = [self.subspace(X[0]) for X in D]
D.sort()
S = Sequence(D, immutable=True, cr=True, universe=int, check=False)
return S
def rational_newforms(self):
"""
Return the newforms with QQ-rational Hecke eigenvalues.
Conjecturally, these correspond to the isogeny classes of
elliptic curves over Q(sqrt(5)) having conductor self.level().
WARNING/TODO: This relies on an unproven (but surely correct)
bound to determine whether a system of Hecke eigenvalues is
really old.
EXAMPLES::
The smallest level example::
sage: from sage.modular.hilbert.sqrt5_hmf import F, QuaternionicModule
sage: H = QuaternionicModule(F.prime_above(31)); D = H.rational_newforms(); D
[
Rational newform number 0...
]
sage: f = D[0]; f
Rational newform number 0 over QQ(sqrt(5)) in Quaternionic module of dimension 2, level 5*a-2 (of norm 31=31) over QQ(sqrt(5))
Notice that computing `a_P` at the bad primes `P` isn't implemented
(it just gives '?')::
sage: f.aplist(50)
[-3, -2, 2, -4, 4, 4, -4, -2, -2, 8, '?', -6, -6, 2]
Another example of higher level::
sage: H = QuaternionicModule(2*F.prime_above(31)); D = H.rational_newforms(); D
[
Rational newform number 0 ...
]
sage: D[0].aplist(33)
['?', 0, -2, 0, -6, 2, 2, 6, 0, -4, '?']
"""
primes = PrimesCoprimeTo(self._level)
D = [X for X in self.new_decomposition() if X.dimension() == 1]
# Have to get rid of the Eisenstein factor
p = primes.next()
while True:
q = p.residue_field().cardinality() + 1
E = [A for A in D if A.hecke_matrix(p)[0,0] == q]
if len(E) == 0:
break
elif len(E) == 1:
D = [A for A in D if A != E[0]]
break
else:
p = primes.next()
Z = []
for number, X in enumerate(D):
f = QuaternionicRationalNewform(X, number)
try:
# ensure that dual eigenspace is defined, i.e., that
# newform really is new.
f.dual_eigenspace()
Z.append(f)
except RuntimeError:
pass
return Sequence(Z, immutable=True, cr=True, universe=int, check=False)
def decomposition(self, B):
"""
Return Hecke decomposition of self using Hecke operators T_p
coprime to the level with norm(p) <= B.
INPUT:
- `B` -- positive integer
OUTPUT:
- sorted Sequence of subspaces of self
EXAMPLES::
sage: from sage.modular.hilbert.sqrt5_hmf import F, QuaternionicModule
sage: H = QuaternionicModule(F.prime_above(31))
sage: H.decomposition(10)
[
Subspace of dimension 1 of Quaternionic module of dimension 2, level 5*a-2 (of norm 31=31) over QQ(sqrt(5)),
Subspace of dimension 1 of Quaternionic module of dimension 2, level 5*a-2 (of norm 31=31) over QQ(sqrt(5))
]
sage: H.decomposition(2)
[
Quaternionic module of dimension 2, level 5*a-2 (of norm 31=31) over QQ(sqrt(5))
]
sage: H = QuaternionicModule(3 * F.prime_above(31)); H
Quaternionic module of dimension 6, level 15*a-6 (of norm 279=3^2*31) over QQ(sqrt(5))
sage: H.decomposition(10)
[
Subspace of dimension 1 ...,
Subspace of dimension 1 ...,
Subspace of dimension 1 ...,
Subspace of dimension 1 ...,
Subspace of dimension 2 ...
]
"""
primes = PrimesCoprimeTo(self.level(), B)
if len(primes) == 0:
D = [self]
else:
T = self.hecke_matrix(primes.next())
D = T.decomposition()
while len([X for X in D if not X[1]]) > 0 and len(primes) > 0:
p = primes.next()
verbose('Norm(p) = %s'%p.norm())
T = self.hecke_matrix(p)
D2 = []
for X in D:
if X[1]:
D2.append(X)
else:
for Z in T.decomposition_of_subspace(X[0]):
D2.append(Z)
D = D2
D = [self.subspace(X[0]) for X in D]
D.sort()
S = Sequence(D, immutable=True, cr=True, universe=int, check=False)
return S
