def cardinality(self):
"""
Returns the number of Lyndon words with the evaluation e.
EXAMPLES::
sage: LyndonWords([]).cardinality()
0
sage: LyndonWords([2,2]).cardinality()
1
sage: LyndonWords([2,3,2]).cardinality()
30
Check to make sure that the count matches up with the number of
Lyndon words generated.
::
sage: comps = [[],[2,2],[3,2,7],[4,2]]+Compositions(4).list()
sage: lws = [ LyndonWords(comp) for comp in comps]
sage: all( [ lw.cardinality() == len(lw.list()) for lw in lws] )
True
"""
evaluation = self.e
le = __builtin__.list(evaluation)
if len(evaluation) == 0:
return 0
n = sum(evaluation)
return sum([
moebius(j) * factorial(n / j) /
prod([factorial(ni / j) for ni in evaluation])
for j in divisors(gcd(le))
]) / n
def _cycle_type(self, s):
"""
EXAMPLES::
sage: cis = species.PartitionSpecies().cycle_index_series()
sage: [cis._cycle_type(p) for p in Partitions(3)]
[[3, 1, 1], [2, 1, 1, 1], [1, 1, 1, 1, 1]]
sage: cis = species.PermutationSpecies().cycle_index_series()
sage: [cis._cycle_type(p) for p in Partitions(3)]
[[3, 1, 1, 1], [2, 2, 1, 1], [1, 1, 1, 1, 1, 1]]
sage: cis = species.SetSpecies().cycle_index_series()
sage: [cis._cycle_type(p) for p in Partitions(3)]
[[1], [1], [1]]
"""
if s == []:
return self._card(0)
res = []
for k in range(1, self._upper_bound_for_longest_cycle(s) + 1):
e = 0
for d in divisors(k):
m = moebius(d)
if m == 0:
continue
u = s.power(k / d)
e += m * self.count(u)
res.extend([k] * int(e / k))
res.reverse()
return Partition(res)
def possible_orders(self):
"""
Return the possible orders of this torsion subgroup, computed from
a known divisor and multiple of the order.
EXAMPLES::
sage: J0(11).rational_torsion_subgroup().possible_orders()
[5]
sage: J0(33).rational_torsion_subgroup().possible_orders()
[100, 200]
Note that this function has not been implemented for `J_1(N)`,
though it should be reasonably easy to do so soon (see Conrad,
Edixhoven, Stein)::
sage: J1(13).rational_torsion_subgroup().possible_orders()
Traceback (most recent call last):
...
NotImplementedError: torsion multiple only implemented for Gamma0
"""
try:
return self._possible_orders
except AttributeError:
pass
u = self.multiple_of_order()
l = self.divisor_of_order()
assert u % l == 0
O = [l * d for d in divisors(u // l)]
self._possible_orders = O
return O
def possible_orders(self):
"""
Return the possible orders of this torsion subgroup, computed from
a known divisor and multiple of the order.
EXAMPLES::
sage: J0(11).rational_torsion_subgroup().possible_orders()
[5]
sage: J0(33).rational_torsion_subgroup().possible_orders()
[100, 200]
Note that this function has not been implemented for `J_1(N)`,
though it should be reasonably easy to do so soon (see Conrad,
Edixhoven, Stein)::
sage: J1(13).rational_torsion_subgroup().possible_orders()
Traceback (most recent call last):
...
NotImplementedError: torsion multiple only implemented for Gamma0
"""
try:
return self._possible_orders
except AttributeError:
pass
u = self.multiple_of_order()
l = self.divisor_of_order()
assert u % l == 0
O = [l * d for d in divisors(u//l)]
self._possible_orders = O
return O
def _cycle_type(self, s):
"""
EXAMPLES::
sage: cis = species.PartitionSpecies().cycle_index_series()
sage: [cis._cycle_type(p) for p in Partitions(3)]
[[3, 1, 1], [2, 1, 1, 1], [1, 1, 1, 1, 1]]
sage: cis = species.PermutationSpecies().cycle_index_series()
sage: [cis._cycle_type(p) for p in Partitions(3)]
[[3, 1, 1, 1], [2, 2, 1, 1], [1, 1, 1, 1, 1, 1]]
sage: cis = species.SetSpecies().cycle_index_series()
sage: [cis._cycle_type(p) for p in Partitions(3)]
[[1], [1], [1]]
"""
if s == []:
return self._card(0)
res = []
for k in range(1, self._upper_bound_for_longest_cycle(s)+1):
e = 0
for d in divisors(k):
m = moebius(d)
if m == 0:
continue
u = s.power(k/d)
e += m*self.count(u)
res.extend([k]*int(e/k))
res.reverse()
return Partition(res)
def _cl_term(n, R=RationalField()):
"""
Compute the order-n term of the cycle index series of the virtual species `\Omega`,
the compositional inverse of the species `E^{+}` of nonempty sets.
EXAMPLES::
sage: from sage.combinat.species.combinatorial_logarithm import _cl_term
sage: [_cl_term(i) for i in range(4)]
[0, p[1], -1/2*p[1, 1] - 1/2*p[2], 1/3*p[1, 1, 1] - 1/3*p[3]]
"""
n = Integer(n) #check that n is an integer
p = SymmetricFunctions(R).power()
res = p.zero()
if n == 1:
res = p([1])
elif n > 1:
res = 1 / n * ((-1)**(n - 1) * p([1])**n -
sum(d * p([Integer(n / d)]).plethysm(_cl_term(d, R))
for d in divisors(n)[:-1]))
return res
def arith_prod_coeff(n):
if n == 0:
res = p.zero()
else:
index_set = ((d, n // d) for d in divisors(n))
res = sum(arith_prod_sf(self.coefficient(i), g.coefficient(j)) for i,j in index_set)
# Build a list which has res in the `n`th slot and 0's before and after
# to feed to sum_generator
res_in_seq = [p.zero()]*n + [res, p.zero()]
return self.parent(res_in_seq)
def arith_prod_coeff(n):
if n == 0:
res = p.zero()
else:
index_set = ((d, n // d) for d in divisors(n))
res = sum(arith_prod_sf(self.coefficient(i), g.coefficient(j)) for i,j in index_set)
# Build a list which has res in the `n`th slot and 0's before and after
# to feed to sum_generator
res_in_seq = [p.zero()]*n + [res, p.zero()]
return self.parent(res_in_seq)
def cardinality(self):
"""
TESTS::
sage: [ LyndonWords(3,i).cardinality() for i in range(1, 11) ]
[3, 3, 8, 18, 48, 116, 312, 810, 2184, 5880]
"""
if self.k == 0:
return 1
else:
s = 0
for d in divisors(self.k):
s += moebius(d)*(self.n**(self.k/d))
return s/self.k
def cardinality(self):
"""
TESTS::
sage: [ LyndonWords(3,i).cardinality() for i in range(1, 11) ]
[3, 3, 8, 18, 48, 116, 312, 810, 2184, 5880]
"""
if self.k == 0:
return 1
else:
s = 0
for d in divisors(self.k):
s += moebius(d) * (self.n**(self.k / d))
return s / self.k
def pAdicEisensteinSeries(self, ring, prec=20):
r"""
Calculate the q-expansion of the p-adic Eisenstein series of given
weight-character, normalised so the constant term is 1.
EXAMPLE::
sage: kappa = pAdicWeightSpace(3)(3, DirichletGroup(3,QQ).0)
sage: kappa.pAdicEisensteinSeries(QQ[['q']], 20)
1 - 9*q + 27*q^2 - 9*q^3 - 117*q^4 + 216*q^5 + 27*q^6 - 450*q^7 + 459*q^8 - 9*q^9 - 648*q^10 + 1080*q^11 - 117*q^12 - 1530*q^13 + 1350*q^14 + 216*q^15 - 1845*q^16 + 2592*q^17 + 27*q^18 - 3258*q^19 + O(q^20)
"""
if not self.is_even():
raise ValueError, "Eisenstein series not defined for odd weight-characters"
q = ring.gen()
s = ring(1) + 2*self.one_over_Lvalue() * sum([sum([self(d)/d for d in divisors(n)]) * q**n for n in xrange(1, prec)])
return s.add_bigoh(prec)
def pAdicEisensteinSeries(self, ring, prec=20):
r"""
Calculate the q-expansion of the p-adic Eisenstein series of given
weight-character, normalised so the constant term is 1.
EXAMPLE::
sage: kappa = pAdicWeightSpace(3)(3, DirichletGroup(3,QQ).0)
sage: kappa.pAdicEisensteinSeries(QQ[['q']], 20)
1 - 9*q + 27*q^2 - 9*q^3 - 117*q^4 + 216*q^5 + 27*q^6 - 450*q^7 + 459*q^8 - 9*q^9 - 648*q^10 + 1080*q^11 - 117*q^12 - 1530*q^13 + 1350*q^14 + 216*q^15 - 1845*q^16 + 2592*q^17 + 27*q^18 - 3258*q^19 + O(q^20)
"""
if not self.is_even():
raise ValueError, "Eisenstein series not defined for odd weight-characters"
q = ring.gen()
s = ring(1) + 2*self.one_over_Lvalue() * sum([sum([self(d)/d for d in divisors(n)]) * q**n for n in xrange(1, prec)])
return s.add_bigoh(prec)
def __iter__(self):
r"""
Iterate over ``self``.
EXAMPLES::
sage: PTC = PrimarySimilarityClassTypes(2)
sage: PTC.cardinality()
3
"""
n = self._n
if self._min[0].divides(n):
for par in Partitions(ZZ(n / self._min[0]), starting=self._min[1]):
yield self.element_class(self, self._min[0], par)
for d in filter(lambda d: d > self._min[0], divisors(n)):
for par in Partitions(ZZ(n / d)):
yield self.element_class(self, d, par)
def __iter__(self):
r"""
Iterate over ``self``.
EXAMPLES::
sage: PTC = PrimarySimilarityClassTypes(2)
sage: PTC.cardinality()
3
"""
n = self._n
if self._min[0].divides(n):
for par in Partitions(ZZ(n / self._min[0]), starting=self._min[1]):
yield self.element_class(self, self._min[0], par)
for d in filter(lambda d: d > self._min[0], divisors(n)):
for par in Partitions(ZZ(n / d)):
yield self.element_class(self, d, par)
def primitives(n, invertible=False, q=None):
"""
Return the number of similarity classes of simple matrices
of order n with entries in a finite field of order ``q``.
This is the same as the number of irreducible polynomials
of degree d.
If ``invertible`` is ``True``, then only the number of
similarity classes of invertible matrices is returned.
.. NOTE::
All primitive classes are invertible unless ``n`` is `1`.
INPUT:
- ``n`` -- a positive integer
- ``invertible`` -- boolean; if set, only number of non-zero classes is returned
- ``q`` -- an integer or an indeterminate
OUTPUT:
- a rational function of the variable ``q``
EXAMPLES::
sage: from sage.combinat.similarity_class_type import primitives
sage: primitives(1)
q
sage: primitives(1, invertible = True)
q - 1
sage: primitives(4)
1/4*q^4 - 1/4*q^2
sage: primitives(4, invertible = True)
1/4*q^4 - 1/4*q^2
"""
if q is None:
q = QQ['q'].gen()
p = sum([moebius(n / d) * q**d for d in divisors(n)]) / n
if invertible and n == 1:
return p - 1
else:
return p
def primitives(n, invertible=False, q=None):
"""
Return the number of similarity classes of simple matrices
of order n with entries in a finite field of order ``q``.
This is the same as the number of irreducible polynomials
of degree d.
If ``invertible`` is ``True``, then only the number of
similarity classes of invertible matrices is returned.
.. NOTE::
All primitive classes are invertible unless ``n`` is `1`.
INPUT:
- ``n`` -- a positive integer
- ``invertible`` -- boolean; if set, only number of non-zero classes is returned
- ``q`` -- an integer or an indeterminate
OUTPUT:
- a rational function of the variable ``q``
EXAMPLES::
sage: from sage.combinat.similarity_class_type import primitives
sage: primitives(1)
q
sage: primitives(1, invertible = True)
q - 1
sage: primitives(4)
1/4*q^4 - 1/4*q^2
sage: primitives(4, invertible = True)
1/4*q^4 - 1/4*q^2
"""
if q is None:
q = QQ["q"].gen()
p = sum([moebius(n / d) * q ** d for d in divisors(n)]) / n
if invertible and n == 1:
return p - 1
else:
return p
def _cis_iterator(self, base_ring):
r"""
The cycle index series of the species of cyclic permutations is
given by
.. math::
-\sum_{k=1}^\infty \phi(k)/k * log(1 - x_k)
which is equal to
.. math::
\sum_{n=1}^\infty \frac{1}{n} * \sum_{k|n} \phi(k) * x_k^{n/k}
.
EXAMPLES::
sage: P = species.CycleSpecies()
sage: cis = P.cycle_index_series()
sage: cis.coefficients(7)
[0,
p[1],
1/2*p[1, 1] + 1/2*p[2],
1/3*p[1, 1, 1] + 2/3*p[3],
1/4*p[1, 1, 1, 1] + 1/4*p[2, 2] + 1/2*p[4],
1/5*p[1, 1, 1, 1, 1] + 4/5*p[5],
1/6*p[1, 1, 1, 1, 1, 1] + 1/6*p[2, 2, 2] + 1/3*p[3, 3] + 1/3*p[6]]
"""
from sage.combinat.sf.sf import SymmetricFunctions
p = SymmetricFunctions(base_ring).power()
zero = base_ring(0)
yield zero
for n in _integers_from(1):
res = zero
for k in divisors(n):
res += euler_phi(k) * p([k]) ** (n // k)
res /= n
yield self._weight * res
def _cis_iterator(self, base_ring):
r"""
The cycle index series of the species of cyclic permutations is
given by
.. math::
-\sum_{k=1}^\infty \phi(k)/k * log(1 - x_k)
which is equal to
.. math::
\sum_{n=1}^\infty \frac{1}{n} * \sum_{k|n} \phi(k) * x_k^{n/k}
.
EXAMPLES::
sage: P = species.CycleSpecies()
sage: cis = P.cycle_index_series()
sage: cis.coefficients(7)
[0,
p[1],
1/2*p[1, 1] + 1/2*p[2],
1/3*p[1, 1, 1] + 2/3*p[3],
1/4*p[1, 1, 1, 1] + 1/4*p[2, 2] + 1/2*p[4],
1/5*p[1, 1, 1, 1, 1] + 4/5*p[5],
1/6*p[1, 1, 1, 1, 1, 1] + 1/6*p[2, 2, 2] + 1/3*p[3, 3] + 1/3*p[6]]
"""
from sage.combinat.sf.all import SFAPower
p = SFAPower(base_ring)
zero = base_ring(0)
yield zero
for n in _integers_from(1):
res = zero
for k in divisors(n):
res += euler_phi(k)*p([k])**(n//k)
res /= n
yield self._weight*res
def cardinality(self):
"""
Returns the number of Lyndon words with the evaluation e.
EXAMPLES::
sage: LyndonWords([]).cardinality()
0
sage: LyndonWords([2,2]).cardinality()
1
sage: LyndonWords([2,3,2]).cardinality()
30
Check to make sure that the count matches up with the number of
Lyndon words generated.
::
sage: comps = [[],[2,2],[3,2,7],[4,2]]+Compositions(4).list()
sage: lws = [ LyndonWords(comp) for comp in comps]
sage: all( [ lw.cardinality() == len(lw.list()) for lw in lws] )
True
"""
evaluation = self.e
le = __builtin__.list(evaluation)
if len(evaluation) == 0:
return 0
n = sum(evaluation)
return (
sum(
[
moebius(j) * factorial(n / j) / prod([factorial(ni / j) for ni in evaluation])
for j in divisors(gcd(le))
]
)
/ n
)
def hecke_operator_on_qexp(f, n, k, eps=None, prec=None, check=True, _return_list=False):
r"""
Given the `q`-expansion `f` of a modular form with character
`\varepsilon`, this function computes the image of `f` under the
Hecke operator `T_{n,k}` of weight `k`.
EXAMPLES::
sage: M = ModularForms(1,12)
sage: hecke_operator_on_qexp(M.basis()[0], 3, 12)
252*q - 6048*q^2 + 63504*q^3 - 370944*q^4 + O(q^5)
sage: hecke_operator_on_qexp(M.basis()[0], 1, 12, prec=7)
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 + O(q^7)
sage: hecke_operator_on_qexp(M.basis()[0], 1, 12)
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + 84480*q^8 - 113643*q^9 - 115920*q^10 + 534612*q^11 - 370944*q^12 - 577738*q^13 + O(q^14)
sage: M.prec(20)
20
sage: hecke_operator_on_qexp(M.basis()[0], 3, 12)
252*q - 6048*q^2 + 63504*q^3 - 370944*q^4 + 1217160*q^5 - 1524096*q^6 + O(q^7)
sage: hecke_operator_on_qexp(M.basis()[0], 1, 12)
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + 84480*q^8 - 113643*q^9 - 115920*q^10 + 534612*q^11 - 370944*q^12 - 577738*q^13 + 401856*q^14 + 1217160*q^15 + 987136*q^16 - 6905934*q^17 + 2727432*q^18 + 10661420*q^19 - 7109760*q^20 + O(q^21)
sage: (hecke_operator_on_qexp(M.basis()[0], 1, 12)*252).add_bigoh(7)
252*q - 6048*q^2 + 63504*q^3 - 370944*q^4 + 1217160*q^5 - 1524096*q^6 + O(q^7)
sage: hecke_operator_on_qexp(M.basis()[0], 6, 12)
-6048*q + 145152*q^2 - 1524096*q^3 + O(q^4)
An example on a formal power series::
sage: R.<q> = QQ[[]]
sage: f = q + q^2 + q^3 + q^7 + O(q^8)
sage: hecke_operator_on_qexp(f, 3, 12)
q + O(q^3)
sage: hecke_operator_on_qexp(delta_qexp(24), 3, 12).prec()
8
sage: hecke_operator_on_qexp(delta_qexp(25), 3, 12).prec()
9
An example of computing `T_{p,k}` in characteristic `p`::
sage: p = 199
sage: fp = delta_qexp(prec=p^2+1, K=GF(p))
sage: tfp = hecke_operator_on_qexp(fp, p, 12)
sage: tfp == fp[p] * fp
True
sage: tf = hecke_operator_on_qexp(delta_qexp(prec=p^2+1), p, 12).change_ring(GF(p))
sage: tfp == tf
True
"""
if eps is None:
# Need to have base_ring=ZZ to work over finite fields, since
# ZZ can coerce to GF(p), but QQ can't.
eps = DirichletGroup(1, base_ring=ZZ)[0]
if check:
if not (is_PowerSeries(f) or is_ModularFormElement(f)):
raise TypeError("f (=%s) must be a power series or modular form" % f)
if not is_DirichletCharacter(eps):
raise TypeError("eps (=%s) must be a Dirichlet character" % eps)
k = Integer(k)
n = Integer(n)
v = []
if prec is None:
if is_ModularFormElement(f):
# always want at least three coefficients, but not too many, unless
# requested
pr = max(f.prec(), f.parent().prec(), (n + 1) * 3)
pr = min(pr, 100 * (n + 1))
prec = pr // n + 1
else:
prec = (f.prec() / ZZ(n)).ceil()
if prec == Infinity:
prec = f.parent().default_prec() // n + 1
if f.prec() < prec:
f._compute_q_expansion(prec)
p = Integer(f.base_ring().characteristic())
if k != 1 and p.is_prime() and n.is_power_of(p):
# if computing T_{p^a} in characteristic p, use the simpler (and faster)
# formula
v = [f[m * n] for m in range(prec)]
else:
l = k - 1
for m in range(prec):
am = sum([eps(d) * d ** l * f[m * n // (d * d)] for d in divisors(gcd(n, m)) if (m * n) % (d * d) == 0])
v.append(am)
if _return_list:
return v
if is_ModularFormElement(f):
R = f.parent()._q_expansion_ring()
else:
R = f.parent()
return R(v, prec)
def __find_eisen_chars_gamma1(N, k):
"""
Find all triples `(\psi_1, \psi_2, t)` that give rise to an Eisenstein series of weight `k` on
`\Gamma_1(N)`.
EXAMPLES::
sage: pars = sage.modular.modform.eis_series.__find_eisen_chars_gamma1(12, 4)
sage: [(x[0].values_on_gens(), x[1].values_on_gens(), x[2]) for x in pars]
[((1, 1), (1, 1), 1),
((1, 1), (1, 1), 2),
((1, 1), (1, 1), 3),
((1, 1), (1, 1), 4),
((1, 1), (1, 1), 6),
((1, 1), (1, 1), 12),
((1, 1), (-1, -1), 1),
((-1, -1), (1, 1), 1),
((-1, 1), (1, -1), 1),
((1, -1), (-1, 1), 1)]
sage: pars = sage.modular.modform.eis_series.__find_eisen_chars_gamma1(12, 5)
sage: [(x[0].values_on_gens(), x[1].values_on_gens(), x[2]) for x in pars]
[((1, 1), (-1, 1), 1),
((1, 1), (-1, 1), 3),
((-1, 1), (1, 1), 1),
((-1, 1), (1, 1), 3),
((1, 1), (1, -1), 1),
((1, 1), (1, -1), 2),
((1, 1), (1, -1), 4),
((1, -1), (1, 1), 1),
((1, -1), (1, 1), 2),
((1, -1), (1, 1), 4)]
"""
pairs = []
s = (-1)**k
G = dirichlet.DirichletGroup(N)
E = list(G)
parity = [c(-1) for c in E]
for i in range(len(E)):
for j in range(i,len(E)):
if parity[i]*parity[j] == s and N % (E[i].conductor()*E[j].conductor()) == 0:
chi, psi = __common_minimal_basering(E[i], E[j])
if k != 1:
pairs.append((chi, psi))
if i!=j: pairs.append((psi,chi))
else:
# if weight is 1 then (chi, psi) and (chi, psi) are the
# same form
if psi.is_trivial() and not chi.is_trivial():
# need to put the trivial character first to get the L-value right
pairs.append((psi, chi))
else:
pairs.append((chi, psi))
#end fors
#end if
triples = []
D = divisors(N)
for chi, psi in pairs:
c_chi = chi.conductor()
c_psi = psi.conductor()
D = divisors(N/(c_chi * c_psi))
if (k==2 and chi.is_trivial() and psi.is_trivial()):
D.remove(1)
chi, psi = __common_minimal_basering(chi, psi)
for t in D:
triples.append((chi, psi, t))
return triples
def __find_eisen_chars(character, k):
"""
Find all triples `(\psi_1, \psi_2, t)` that give rise to an Eisenstein series of the given weight and character.
EXAMPLES::
sage: sage.modular.modform.eis_series.__find_eisen_chars(DirichletGroup(36).0, 4)
[]
sage: pars = sage.modular.modform.eis_series.__find_eisen_chars(DirichletGroup(36).0, 5)
sage: [(x[0].values_on_gens(), x[1].values_on_gens(), x[2]) for x in pars]
[((1, 1), (-1, 1), 1),
((1, 1), (-1, 1), 3),
((1, 1), (-1, 1), 9),
((1, -1), (-1, -1), 1),
((-1, 1), (1, 1), 1),
((-1, 1), (1, 1), 3),
((-1, 1), (1, 1), 9),
((-1, -1), (1, -1), 1)]
"""
N = character.modulus()
if character.is_trivial():
if k%2 != 0:
return []
char_inv = ~character
V = [(character, char_inv, t) for t in divisors(N) if t>1]
if k != 2:
V.insert(0,(character, char_inv, 1))
if is_squarefree(N):
return V
# Now include all pairs (chi,chi^(-1)) such that cond(chi)^2 divides N:
# TODO: Optimize -- this is presumably way too hard work below.
G = dirichlet.DirichletGroup(N)
for chi in G:
if not chi.is_trivial():
f = chi.conductor()
if N % (f**2) == 0:
chi = chi.minimize_base_ring()
chi_inv = ~chi
for t in divisors(N//(f**2)):
V.insert(0, (chi, chi_inv, t))
return V
eps = character
if eps(-1) != (-1)**k:
return []
eps = eps.maximize_base_ring()
G = eps.parent()
# Find all pairs chi, psi such that:
#
# (1) cond(chi)*cond(psi) divides the level, and
#
# (2) chi*psi == eps, where eps is the nebentypus character of self.
#
# See [Miyake, Modular Forms] Lemma 7.1.1.
K = G.base_ring()
C = {}
t0 = misc.cputime()
for e in G:
m = Integer(e.conductor())
if m in C:
C[m].append(e)
else:
C[m] = [e]
misc.verbose("Enumeration with conductors.",t0)
params = []
for L in divisors(N):
misc.verbose("divisor %s"%L)
if L not in C:
continue
GL = C[L]
for R in divisors(N/L):
if R not in C:
continue
GR = C[R]
for chi in GL:
for psi in GR:
if chi*psi == eps:
chi0, psi0 = __common_minimal_basering(chi, psi)
for t in divisors(N//(R*L)):
if k != 1 or ((psi0, chi0, t) not in params):
params.append( (chi0,psi0,t) )
return params
def dimension_new_cusp_forms(self, k=2, eps=None, p=0, algorithm="CohenOesterle"):
r"""
Dimension of the new subspace (or `p`-new subspace) of cusp forms of
weight `k` and character `\varepsilon`.
INPUT:
- ``k`` - an integer (default: 2)
- ``eps`` - a Dirichlet character
- ``p`` - a prime (default: 0); just the `p`-new subspace if given
- ``algorithm`` - either "CohenOesterle" (the default) or "Quer". This
specifies the method to use in the case of nontrivial character:
either the Cohen--Oesterle formula as described in Stein's book, or
by Moebius inversion using the subgroups GammaH (a method due to
Jordi Quer).
EXAMPLES::
sage: G = DirichletGroup(9)
sage: eps = G.0^3
sage: eps.conductor()
3
sage: [Gamma1(9).dimension_new_cusp_forms(k, eps) for k in [2..10]]
[0, 0, 0, 2, 0, 2, 0, 2, 0]
sage: [Gamma1(9).dimension_cusp_forms(k, eps) for k in [2..10]]
[0, 0, 0, 2, 0, 4, 0, 6, 0]
sage: [Gamma1(9).dimension_new_cusp_forms(k, eps, 3) for k in [2..10]]
[0, 0, 0, 2, 0, 2, 0, 2, 0]
Double check using modular symbols (independent calculation)::
sage: [ModularSymbols(eps,k,sign=1).cuspidal_subspace().new_subspace().dimension() for k in [2..10]]
[0, 0, 0, 2, 0, 2, 0, 2, 0]
sage: [ModularSymbols(eps,k,sign=1).cuspidal_subspace().new_subspace(3).dimension() for k in [2..10]]
[0, 0, 0, 2, 0, 2, 0, 2, 0]
Another example at level 33::
sage: G = DirichletGroup(33)
sage: eps = G.1
sage: eps.conductor()
11
sage: [Gamma1(33).dimension_new_cusp_forms(k, G.1) for k in [2..4]]
[0, 4, 0]
sage: [Gamma1(33).dimension_new_cusp_forms(k, G.1, algorithm="Quer") for k in [2..4]]
[0, 4, 0]
sage: [Gamma1(33).dimension_new_cusp_forms(k, G.1^2) for k in [2..4]]
[2, 0, 6]
sage: [Gamma1(33).dimension_new_cusp_forms(k, G.1^2, p=3) for k in [2..4]]
[2, 0, 6]
"""
if eps == None:
return GammaH_class.dimension_new_cusp_forms(self, k, p)
N = self.level()
eps = DirichletGroup(N)(eps)
from all import Gamma0
if eps.is_trivial():
return Gamma0(N).dimension_new_cusp_forms(k, p)
from congroup_gammaH import mumu
if p == 0 or N%p != 0 or eps.conductor().valuation(p) == N.valuation(p):
D = [eps.conductor()*d for d in divisors(N//eps.conductor())]
return sum([Gamma1_constructor(M).dimension_cusp_forms(k, eps.restrict(M), algorithm)*mumu(N//M) for M in D])
eps_p = eps.restrict(N//p)
old = Gamma1_constructor(N//p).dimension_cusp_forms(k, eps_p, algorithm)
return self.dimension_cusp_forms(k, eps, algorithm) - 2*old
def hecke_operator_on_qexp(f,
n,
k,
eps=None,
prec=None,
check=True,
_return_list=False):
r"""
Given the `q`-expansion `f` of a modular form with character
`\varepsilon`, this function computes the image of `f` under the
Hecke operator `T_{n,k}` of weight `k`.
EXAMPLES::
sage: M = ModularForms(1,12)
sage: hecke_operator_on_qexp(M.basis()[0], 3, 12)
252*q - 6048*q^2 + 63504*q^3 - 370944*q^4 + O(q^5)
sage: hecke_operator_on_qexp(M.basis()[0], 1, 12, prec=7)
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 + O(q^7)
sage: hecke_operator_on_qexp(M.basis()[0], 1, 12)
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + 84480*q^8 - 113643*q^9 - 115920*q^10 + 534612*q^11 - 370944*q^12 - 577738*q^13 + O(q^14)
sage: M.prec(20)
20
sage: hecke_operator_on_qexp(M.basis()[0], 3, 12)
252*q - 6048*q^2 + 63504*q^3 - 370944*q^4 + 1217160*q^5 - 1524096*q^6 + O(q^7)
sage: hecke_operator_on_qexp(M.basis()[0], 1, 12)
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + 84480*q^8 - 113643*q^9 - 115920*q^10 + 534612*q^11 - 370944*q^12 - 577738*q^13 + 401856*q^14 + 1217160*q^15 + 987136*q^16 - 6905934*q^17 + 2727432*q^18 + 10661420*q^19 - 7109760*q^20 + O(q^21)
sage: (hecke_operator_on_qexp(M.basis()[0], 1, 12)*252).add_bigoh(7)
252*q - 6048*q^2 + 63504*q^3 - 370944*q^4 + 1217160*q^5 - 1524096*q^6 + O(q^7)
sage: hecke_operator_on_qexp(M.basis()[0], 6, 12)
-6048*q + 145152*q^2 - 1524096*q^3 + O(q^4)
An example on a formal power series::
sage: R.<q> = QQ[[]]
sage: f = q + q^2 + q^3 + q^7 + O(q^8)
sage: hecke_operator_on_qexp(f, 3, 12)
q + O(q^3)
sage: hecke_operator_on_qexp(delta_qexp(24), 3, 12).prec()
8
sage: hecke_operator_on_qexp(delta_qexp(25), 3, 12).prec()
9
An example of computing `T_{p,k}` in characteristic `p`::
sage: p = 199
sage: fp = delta_qexp(prec=p^2+1, K=GF(p))
sage: tfp = hecke_operator_on_qexp(fp, p, 12)
sage: tfp == fp[p] * fp
True
sage: tf = hecke_operator_on_qexp(delta_qexp(prec=p^2+1), p, 12).change_ring(GF(p))
sage: tfp == tf
True
"""
if eps is None:
# Need to have base_ring=ZZ to work over finite fields, since
# ZZ can coerce to GF(p), but QQ can't.
eps = DirichletGroup(1, base_ring=ZZ).gen(0)
if check:
if not (is_PowerSeries(f) or is_ModularFormElement(f)):
raise TypeError, "f (=%s) must be a power series or modular form" % f
if not is_DirichletCharacter(eps):
raise TypeError, "eps (=%s) must be a Dirichlet character" % eps
k = Integer(k)
n = Integer(n)
v = []
if prec is None:
if is_ModularFormElement(f):
# always want at least three coefficients, but not too many, unless
# requested
pr = max(f.prec(), f.parent().prec(), (n + 1) * 3)
pr = min(pr, 100 * (n + 1))
prec = pr // n + 1
else:
prec = (f.prec() / ZZ(n)).ceil()
if prec == Infinity: prec = f.parent().default_prec() // n + 1
if f.prec() < prec:
f._compute_q_expansion(prec)
p = Integer(f.base_ring().characteristic())
if k != 1 and p.is_prime() and n.is_power_of(p):
# if computing T_{p^a} in characteristic p, use the simpler (and faster)
# formula
v = [f[m * n] for m in range(prec)]
else:
l = k - 1
for m in range(prec):
am = sum([eps(d) * d**l * f[m*n//(d*d)] for \
d in divisors(gcd(n, m)) if (m*n) % (d*d) == 0])
v.append(am)
if _return_list:
return v
if is_ModularFormElement(f):
R = f.parent()._q_expansion_ring()
else:
R = f.parent()
return R(v, prec)
def __find_eisen_chars_gamma1(N, k):
"""
Find all triples `(\psi_1, \psi_2, t)` that give rise to an Eisenstein series of weight `k` on
`\Gamma_1(N)`.
EXAMPLES::
sage: pars = sage.modular.modform.eis_series.__find_eisen_chars_gamma1(12, 4)
sage: [(x[0].values_on_gens(), x[1].values_on_gens(), x[2]) for x in pars]
[((1, 1), (1, 1), 1),
((1, 1), (1, 1), 2),
((1, 1), (1, 1), 3),
((1, 1), (1, 1), 4),
((1, 1), (1, 1), 6),
((1, 1), (1, 1), 12),
((1, 1), (-1, -1), 1),
((-1, -1), (1, 1), 1),
((-1, 1), (1, -1), 1),
((1, -1), (-1, 1), 1)]
sage: pars = sage.modular.modform.eis_series.__find_eisen_chars_gamma1(12, 5)
sage: [(x[0].values_on_gens(), x[1].values_on_gens(), x[2]) for x in pars]
[((1, 1), (-1, 1), 1),
((1, 1), (-1, 1), 3),
((-1, 1), (1, 1), 1),
((-1, 1), (1, 1), 3),
((1, 1), (1, -1), 1),
((1, 1), (1, -1), 2),
((1, 1), (1, -1), 4),
((1, -1), (1, 1), 1),
((1, -1), (1, 1), 2),
((1, -1), (1, 1), 4)]
"""
pairs = []
s = (-1)**k
G = dirichlet.DirichletGroup(N)
E = list(G)
parity = [c(-1) for c in E]
for i in range(len(E)):
for j in range(i, len(E)):
if parity[i] * parity[j] == s and N % (E[i].conductor() *
E[j].conductor()) == 0:
chi, psi = __common_minimal_basering(E[i], E[j])
if k != 1:
pairs.append((chi, psi))
if i != j: pairs.append((psi, chi))
else:
# if weight is 1 then (chi, psi) and (chi, psi) are the
# same form
if psi.is_trivial() and not chi.is_trivial():
# need to put the trivial character first to get the L-value right
pairs.append((psi, chi))
else:
pairs.append((chi, psi))
#end fors
#end if
triples = []
D = divisors(N)
for chi, psi in pairs:
c_chi = chi.conductor()
c_psi = psi.conductor()
D = divisors(N / (c_chi * c_psi))
if (k == 2 and chi.is_trivial() and psi.is_trivial()):
D.remove(1)
chi, psi = __common_minimal_basering(chi, psi)
for t in D:
triples.append((chi, psi, t))
return triples
def __find_eisen_chars(character, k):
"""
Find all triples `(\psi_1, \psi_2, t)` that give rise to an Eisenstein series of the given weight and character.
EXAMPLES::
sage: sage.modular.modform.eis_series.__find_eisen_chars(DirichletGroup(36).0, 4)
[]
sage: pars = sage.modular.modform.eis_series.__find_eisen_chars(DirichletGroup(36).0, 5)
sage: [(x[0].values_on_gens(), x[1].values_on_gens(), x[2]) for x in pars]
[((1, 1), (-1, 1), 1),
((1, 1), (-1, 1), 3),
((1, 1), (-1, 1), 9),
((1, -1), (-1, -1), 1),
((-1, 1), (1, 1), 1),
((-1, 1), (1, 1), 3),
((-1, 1), (1, 1), 9),
((-1, -1), (1, -1), 1)]
"""
N = character.modulus()
if character.is_trivial():
if k % 2 != 0:
return []
char_inv = ~character
V = [(character, char_inv, t) for t in divisors(N) if t > 1]
if k != 2:
V.insert(0, (character, char_inv, 1))
if is_squarefree(N):
return V
# Now include all pairs (chi,chi^(-1)) such that cond(chi)^2 divides N:
# TODO: Optimize -- this is presumably way too hard work below.
G = dirichlet.DirichletGroup(N)
for chi in G:
if not chi.is_trivial():
f = chi.conductor()
if N % (f**2) == 0:
chi = chi.minimize_base_ring()
chi_inv = ~chi
for t in divisors(N // (f**2)):
V.insert(0, (chi, chi_inv, t))
return V
eps = character
if eps(-1) != (-1)**k:
return []
eps = eps.maximize_base_ring()
G = eps.parent()
# Find all pairs chi, psi such that:
#
# (1) cond(chi)*cond(psi) divides the level, and
#
# (2) chi*psi == eps, where eps is the nebentypus character of self.
#
# See [Miyake, Modular Forms] Lemma 7.1.1.
K = G.base_ring()
C = {}
t0 = misc.cputime()
for e in G:
m = Integer(e.conductor())
if C.has_key(m):
C[m].append(e)
else:
C[m] = [e]
misc.verbose("Enumeration with conductors.", t0)
params = []
for L in divisors(N):
misc.verbose("divisor %s" % L)
if not C.has_key(L):
continue
GL = C[L]
for R in divisors(N / L):
if not C.has_key(R):
continue
GR = C[R]
for chi in GL:
for psi in GR:
if chi * psi == eps:
chi0, psi0 = __common_minimal_basering(chi, psi)
for t in divisors(N // (R * L)):
if k != 1 or ((psi0, chi0, t) not in params):
params.append((chi0, psi0, t))
return params
def dimension_new_cusp_forms(self,
k=2,
eps=None,
p=0,
algorithm="CohenOesterle"):
r"""
Dimension of the new subspace (or `p`-new subspace) of cusp forms of
weight `k` and character `\varepsilon`.
INPUT:
- ``k`` - an integer (default: 2)
- ``eps`` - a Dirichlet character
- ``p`` - a prime (default: 0); just the `p`-new subspace if given
- ``algorithm`` - either "CohenOesterle" (the default) or "Quer". This
specifies the method to use in the case of nontrivial character:
either the Cohen--Oesterle formula as described in Stein's book, or
by Moebius inversion using the subgroups GammaH (a method due to
Jordi Quer).
EXAMPLES::
sage: G = DirichletGroup(9)
sage: eps = G.0^3
sage: eps.conductor()
3
sage: [Gamma1(9).dimension_new_cusp_forms(k, eps) for k in [2..10]]
[0, 0, 0, 2, 0, 2, 0, 2, 0]
sage: [Gamma1(9).dimension_cusp_forms(k, eps) for k in [2..10]]
[0, 0, 0, 2, 0, 4, 0, 6, 0]
sage: [Gamma1(9).dimension_new_cusp_forms(k, eps, 3) for k in [2..10]]
[0, 0, 0, 2, 0, 2, 0, 2, 0]
Double check using modular symbols (independent calculation)::
sage: [ModularSymbols(eps,k,sign=1).cuspidal_subspace().new_subspace().dimension() for k in [2..10]]
[0, 0, 0, 2, 0, 2, 0, 2, 0]
sage: [ModularSymbols(eps,k,sign=1).cuspidal_subspace().new_subspace(3).dimension() for k in [2..10]]
[0, 0, 0, 2, 0, 2, 0, 2, 0]
Another example at level 33::
sage: G = DirichletGroup(33)
sage: eps = G.1
sage: eps.conductor()
11
sage: [Gamma1(33).dimension_new_cusp_forms(k, G.1) for k in [2..4]]
[0, 4, 0]
sage: [Gamma1(33).dimension_new_cusp_forms(k, G.1, algorithm="Quer") for k in [2..4]]
[0, 4, 0]
sage: [Gamma1(33).dimension_new_cusp_forms(k, G.1^2) for k in [2..4]]
[2, 0, 6]
sage: [Gamma1(33).dimension_new_cusp_forms(k, G.1^2, p=3) for k in [2..4]]
[2, 0, 6]
"""
if eps == None:
return GammaH_class.dimension_new_cusp_forms(self, k, p)
N = self.level()
eps = DirichletGroup(N)(eps)
from all import Gamma0
if eps.is_trivial():
return Gamma0(N).dimension_new_cusp_forms(k, p)
from congroup_gammaH import mumu
if p == 0 or N % p != 0 or eps.conductor().valuation(p) == N.valuation(
p):
D = [eps.conductor() * d for d in divisors(N // eps.conductor())]
return sum([
Gamma1_constructor(M).dimension_cusp_forms(
k, eps.restrict(M), algorithm) * mumu(N // M) for M in D
])
eps_p = eps.restrict(N // p)
old = Gamma1_constructor(N // p).dimension_cusp_forms(
k, eps_p, algorithm)
return self.dimension_cusp_forms(k, eps, algorithm) - 2 * old
def _cl_term(n, R = RationalField()):
"""
Compute the order-n term of the cycle index series of the virtual species `\Omega`,
the compositional inverse of the species `E^{+}` of nonempty sets.
EXAMPLES::
sage: from sage.combinat.species.generating_series import _cl_term
sage: [_cl_term(i) for i in range(4)]
[0, p[1], -1/2*p[1, 1] - 1/2*p[2], 1/3*p[1, 1, 1] - 1/3*p[3]]
"""
n = Integer(n) #check that n is an integer
p = SymmetricFunctions(R).power()
res = p.zero()
if n == 1:
res = p([1])
elif n > 1:
res = 1/n * ((-1)**(n-1) * p([1])**n - sum(d * p([Integer(n/d)]).plethysm(_cl_term(d, R)) for d in divisors(n)[:-1]))
return res
