def submodule_generated_by_images(self, M):
"""
Return the submodule of this ambient modular symbols space
generated by the images under all degeneracy maps of M. The space M
must have the same weight, sign, and group or character as this
ambient space.
EXAMPLES::
sage: ModularSymbols(6, 12).submodule_generated_by_images(ModularSymbols(1,12))
Modular Symbols subspace of dimension 12 of Modular Symbols space of dimension 22 for Gamma_0(6) of weight 12 with sign 0 over Rational Field
"""
S = self.zero_submodule()
if self.level() % M.level() == 0:
D = arith.divisors(self.level() // M.level())
elif M.level() % self.level() == 0:
D = arith.divisors(M.level() // self.level())
else:
D = []
for t in D:
d = M.degeneracy_map(self, t)
if d.codomain() != self:
raise ArithmeticError("incompatible spaces of modular symbols")
S += d.image()
if self.is_full_hecke_module(compute=False):
S._is_full_hecke_module = True
return S
def find_product_decomposition(g, k, lmbda=1):
r"""
Try to find a product decomposition construction for difference matrices.
INPUT:
- ``g,k,lmbda`` -- integers, parameters of the difference matrix
OUTPUT:
A pair of pairs ``(g1,lmbda),(g2,lmbda2)`` if Sage knows how to build
`(g1,k,lmbda1)` and `(g2,k,lmbda2)` difference matrices and ``False``
otherwise.
EXAMPLES::
sage: from sage.combinat.designs.difference_matrices import find_product_decomposition
sage: find_product_decomposition(77,6)
((7, 1), (11, 1))
sage: find_product_decomposition(616,7)
((7, 1), (88, 1))
sage: find_product_decomposition(24,10)
False
"""
for lmbda1 in divisors(lmbda):
lmbda2 = lmbda//lmbda1
# To avoid infinite loop:
# if lmbda1 == lmbda, then g1 should not be g
# if lmbda2 == lmbda, then g2 should not be g
if lmbda1 == lmbda:
if lmbda2 == lmbda:
div = divisors(g)[1:-1]
else:
div = divisors(g)[:-1]
else:
if lmbda2 == lmbda:
div = divisors(g)[1:]
else:
div = divisors(g)
for g1 in div:
g2 = g//g1
if g1 > g2:
break
if (difference_matrix(g1,k,lmbda1,existence=True) and
difference_matrix(g2,k,lmbda2,existence=True)):
return (g1,lmbda1),(g2,lmbda2)
return False
def _b_power_k(self, k):
r"""
An expression involving Moebius inversion in the powersum generators.
For a positive value of ``k``, this expression is
.. MATH::
\frac{1}{k} \sum_{d|k} \mu(d/k) p_d.
INPUT:
- ``k`` -- a positive integer
OUTPUT:
- an expression in the powersum basis of the symmetric functions
EXAMPLES::
sage: st = SymmetricFunctions(QQ).st()
sage: st._b_power_k(1)
p[1]
sage: st._b_power_k(2)
-1/2*p[1] + 1/2*p[2]
sage: st._b_power_k(6)
1/6*p[1] - 1/6*p[2] - 1/6*p[3] + 1/6*p[6]
"""
if k == 1:
return self._p([1])
if k > 0:
return ~k * self._p.linear_combination((self._p([d]),moebius(k//d))
for d in divisors(k))
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 number_of_Gamma0_NFCusps(N):
"""
Returns the total number of orbits of cusps under the action of the
congruence subgroup `\\Gamma_0(N)`.
INPUT:
- ``N`` -- a number field ideal.
OUTPUT:
ingeter -- the number of orbits of cusps under Gamma0(N)-action.
EXAMPLES::
sage: k.<a> = NumberField(x^3 + 11)
sage: N = k.ideal(2, a+1)
sage: from sage.modular.cusps_nf import number_of_Gamma0_NFCusps
sage: number_of_Gamma0_NFCusps(N)
4
sage: L = Gamma0_NFCusps(N)
sage: len(L) == number_of_Gamma0_NFCusps(N)
True
"""
k = N.number_field()
# The number of Gamma0(N)-sub-orbits for each Gamma-orbit:
from sage.arith.all import divisors
s = sum([len(list((d+N/d).invertible_residues_mod(k.unit_group().gens()))) \
for d in divisors(N)])
# There are h Gamma-orbits, with h class number of underlying number field.
return s*k.class_number()
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 AllCusps(N):
r"""
Return a list of CuspFamily objects corresponding to the cusps of
`X_0(N)`.
INPUT:
- ``N`` - (integer): the level
EXAMPLES::
sage: AllCusps(18)
[(Inf), (c_{2}), (c_{3,1}), (c_{3,2}), (c_{6,1}), (c_{6,2}), (c_{9}), (0)]
sage: AllCusps(0)
Traceback (most recent call last):
...
ValueError: N must be positive
"""
N = ZZ(N)
if N <= 0:
raise ValueError("N must be positive")
c = []
for d in divisors(N):
n = num_cusps_of_width(N, d)
if n == 1:
c.append(CuspFamily(N, d))
elif n > 1:
for i in xrange(n):
c.append(CuspFamily(N, d, label=str(i+1)))
return c
def _coset_reduction_data_second_coord(G):
"""
Compute data used for determining the canonical coset
representative of an element of SL_2(Z) modulo G. This
function specifically returns data needed for the second part
of the reduction step (the second coordinate).
INPUT:
self
OUTPUT:
a dictionary v with keys the divisors of N such that v[d]
is the subgroup {h in H : h = 1 (mod N/d)}.
EXAMPLES::
sage: G = GammaH(240,[7,239])
sage: G._coset_reduction_data_second_coord()
{1: [1],
2: [1],
3: [1],
4: [1],
5: [1, 49],
6: [1],
8: [1],
10: [1, 49],
12: [1],
15: [1, 49],
16: [1],
20: [1, 49],
24: [1, 191],
30: [1, 49, 137, 233],
40: [1, 7, 49, 103],
48: [1, 191],
60: [1, 49, 137, 233],
80: [1, 7, 49, 103],
120: [1, 7, 49, 103, 137, 191, 233, 239],
240: [1, 7, 49, 103, 137, 191, 233, 239]}
sage: G = GammaH(1200,[-1,7]); G
Congruence Subgroup Gamma_H(1200) with H generated by [7, 1199]
sage: K = G._coset_reduction_data_second_coord().keys() ; K.sort()
sage: K == divisors(1200)
True
"""
H = G._list_of_elements_in_H()
N = G.level()
v = { 1: [1] , N: H }
for d in [x for x in divisors(N) if x > 1 and x < N ]:
N_over_d = N // d
v[d] = [x for x in H if x % N_over_d == 1]
return v
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 Integer(1)
else:
s = Integer(0)
for d in divisors(self._k):
s += moebius(d)*(self._n**(self._k/d))
return s/self._k
def reduce_basis(self, long_etas):
r"""
Produce a more manageable basis via LLL-reduction.
INPUT:
- ``long_etas`` - a list of EtaGroupElement objects (which
should all be of the same level)
OUTPUT:
- a new list of EtaGroupElement objects having
hopefully smaller norm
ALGORITHM: We define the norm of an eta-product to be the
`L^2` norm of its divisor (as an element of the free
`\ZZ`-module with the cusps as basis and the
standard inner product). Applying LLL-reduction to this gives a
basis of hopefully more tractable elements. Of course we'd like to
use the `L^1` norm as this is just twice the degree, which
is a much more natural invariant, but `L^2` norm is easier
to work with!
EXAMPLES::
sage: EtaGroup(4).reduce_basis([ EtaProduct(4, {1:8,2:24,4:-32}), EtaProduct(4, {1:8, 4:-8})])
[Eta product of level 4 : (eta_1)^8 (eta_4)^-8,
Eta product of level 4 : (eta_1)^-8 (eta_2)^24 (eta_4)^-16]
"""
from six.moves import range
N = self.level()
cusps = AllCusps(N)
r = matrix(ZZ, [[et.order_at_cusp(c) for c in cusps] for et in long_etas])
V = FreeModule(ZZ, r.ncols())
A = V.submodule_with_basis([V(rw) for rw in r.rows()])
rred = r.LLL()
short_etas = []
for shortvect in rred.rows():
bv = A.coordinates(shortvect)
dict = {d: sum(bv[i] * long_etas[i].r(d)
for i in range(r.nrows()))
for d in divisors(N)}
short_etas.append(self(dict))
return short_etas
def eval_at_permutation_roots_on_generators(self, k, rho):
r"""
Evaluate `p_k` at eigenvalues of permutation matrix.
This function evaluates a symmetric function ``p([k])``
at the eigenvalues of a permutation matrix with cycle
structure ``\rho``.
This function evaluates a `p_k` at the roots of unity
.. MATH::
\Xi_{\rho_1},\Xi_{\rho_2},\ldots,\Xi_{\rho_\ell}
where
.. MATH::
\Xi_{m} = 1,\zeta_m,\zeta_m^2,\ldots,\zeta_m^{m-1}
and `\zeta_m` is an `m` root of unity.
This is characterized by `p_k[ A , B ] = p_k[A] + p_k[B]` and
`p_k[ \Xi_m ] = 0` unless `m` divides `k` and `p_{rm}[\Xi_m]=m`.
INPUT:
- ``k`` -- a non-negative integer
- ``rho`` -- a partition or a list of non-negative integers
OUTPUT:
- an element of the base ring
EXAMPLES::
sage: p = SymmetricFunctions(QQ).p()
sage: p.eval_at_permutation_roots_on_generators(3, [6])
0
sage: p.eval_at_permutation_roots_on_generators(3, [3])
3
sage: p.eval_at_permutation_roots_on_generators(3, [1])
1
sage: p.eval_at_permutation_roots_on_generators(3, [3,3])
6
sage: p.eval_at_permutation_roots_on_generators(3, [1,1,1,1,1])
5
"""
return self.base_ring().sum(d*list(rho).count(d) for d in divisors(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 cardinality(self):
r"""
Return the number of integer necklaces with the evaluation ``content``.
The formula for the number of necklaces of content `\alpha`
a composition of `n` is:
.. MATH::
\sum_{d|gcd(\alpha)} \phi(d)
\binom{n/d}{\alpha_1/d, \ldots, \alpha_\ell/d},
where `\phi(d)` is the Euler `\phi` function.
EXAMPLES::
sage: Necklaces([]).cardinality()
0
sage: Necklaces([2,2]).cardinality()
2
sage: Necklaces([2,3,2]).cardinality()
30
sage: Necklaces([0,3,2]).cardinality()
2
Check to make sure that the count matches up with the number of
necklace words generated.
::
sage: comps = [[],[2,2],[3,2,7],[4,2],[0,4,2],[2,0,4]]+Compositions(4).list()
sage: ns = [Necklaces(comp) for comp in comps]
sage: all(n.cardinality() == len(n.list()) for n in ns)
True
"""
evaluation = self._content
le = list(evaluation)
if not le:
return ZZ.zero()
n = sum(le)
return ZZ.sum(euler_phi(j) * factorial(n // j) //
prod(factorial(ni // j) for ni in evaluation)
for j in divisors(gcd(le))) // n
def p1NFlist(N):
"""
Returns a list of the normalized elements of `\\mathbb{P}^1(R/N)`, where
`N` is an integral ideal.
INPUT:
- ``N`` - integral ideal (the level or modulus).
EXAMPLES::
sage: k.<a> = NumberField(x^2 + 23)
sage: N = k.ideal(3)
sage: from sage.modular.modsym.p1list_nf import p1NFlist, psi
sage: len(p1NFlist(N))==psi(N)
True
"""
k = N.number_field()
L = [MSymbol(N, k(0),k(1), check=False)]
#N.residues() = iterator through the residues mod N
L = L+[MSymbol(N, k(1), r, check=False) for r in N.residues()]
from sage.arith.all import divisors
for D in divisors(N):
if not D.is_trivial() and D!=N:
#we find Dp ideal coprime to N, in inverse class to D
if D.is_principal():
Dp = k.ideal(1)
c = D.gens_reduced()[0]
else:
it = k.primes_of_degree_one_iter()
Dp = next(it)
while not Dp.is_coprime(N) or not (Dp*D).is_principal():
Dp = next(it)
c = (D*Dp).gens_reduced()[0]
#now we find all the (c,d)'s which have associated divisor D
I = D + N/D
for d in (N/D).residues():
if I.is_coprime(d):
M = D.prime_to_idealM_part(N/D)
u = (Dp*M).element_1_mod(N/D)
d1 = u*d + (1-u)
L.append(MSymbol(N, c, d1, check=False).normalize())
return L
def lift2smallest_field2(a):
"""
INPUT: a is an element of a finite field GF(q)
OUTPUT: the element b of the smallest subfield F of GF(q) for which F(b)=a.
EXAMPLES::
sage: from sage.coding.code_constructions import lift2smallest_field2
sage: FF.<z> = GF(3^4,"z")
sage: a = z^40
sage: lift2smallest_field2(a)
doctest:...: DeprecationWarning: lift2smallest_field2 will be removed in a future release of Sage. Consider using sage.coding.code_constructions._lift2smallest_field instead, though this is private and may be removed in the future without deprecation warning. If you care about this functionality being in Sage, consider opening a Trac ticket for promoting the function to public.
See http://trac.sagemath.org/21165 for details.
(2, Finite Field of size 3)
sage: FF.<z> = GF(2^4,"z")
sage: a = z^15
sage: lift2smallest_field2(a)
(1, Finite Field of size 2)
.. warning::
Since coercion (the FF(b) step) has a bug in it, this
*only works* in the case when you *know* F is a prime field.
AUTHORS:
- David Joyner
"""
deprecation(
21165,
"lift2smallest_field2 will be removed in a future release of Sage. Consider using sage.coding.code_constructions._lift2smallest_field instead, though this is private and may be removed in the future without deprecation warning. If you care about this functionality being in Sage, consider opening a Trac ticket for promoting the function to public.",
)
FF = a.parent()
q = FF.order()
if q.is_prime():
return a, FF
p = q.factor()[0][0]
k = q.factor()[0][1]
for d in divisors(k):
F = GF(p ** d, "zz")
for b in F:
if FF(b) == a:
return b, F
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 _find_cusps(self):
r"""
Return an ordered list of inequivalent cusps for self, i.e. a
set of representatives for the orbits of self on
`\mathbb{P}^1(\QQ)`. These are returned in a reduced
form; see self.reduce_cusp for the definition of reduced.
ALGORITHM:
Uses explicit formulae specific to `\Gamma_0(N)`: a reduced cusp on
`\Gamma_0(N)` is always of the form `a/d` where `d | N`, and `a_1/d
\sim a_2/d` if and only if `a_1 \cong a_2 \bmod {\rm gcd}(d,
N/d)`.
EXAMPLES::
sage: Gamma0(90)._find_cusps()
[0, 1/45, 1/30, 1/18, 1/15, 1/10, 1/9, 2/15, 1/6, 1/5, 1/3, 11/30, 1/2, 2/3, 5/6, Infinity]
sage: Gamma0(1).cusps()
[Infinity]
sage: Gamma0(180).cusps() == Gamma0(180).cusps(algorithm='modsym')
True
"""
N = self.level()
s = []
for d in arith.divisors(N):
w = arith.gcd(d, N//d)
if w == 1:
if d == 1:
s.append(Cusp(1,0))
elif d == N:
s.append(Cusp(0,1))
else:
s.append(Cusp(1,d))
else:
for a in xrange(1, w):
if arith.gcd(a, w) == 1:
while arith.gcd(a, d//w) != 1:
a += w
s.append(Cusp(a,d))
return sorted(s)
def nregcusps(self):
r"""
Return the number of orbits of regular cusps for this subgroup. A cusp is regular
if we may find a parabolic element generating the stabiliser of that
cusp whose eigenvalues are both +1 rather than -1. If G contains -1,
all cusps are regular.
EXAMPLES::
sage: GammaH(20, [17]).nregcusps()
4
sage: GammaH(20, [17]).nirregcusps()
2
sage: GammaH(3212, [2045, 2773]).nregcusps()
1440
sage: GammaH(3212, [2045, 2773]).nirregcusps()
720
AUTHOR:
- Jordi Quer
"""
if self.is_even():
return self.ncusps()
N = self.level()
H = self._list_of_elements_in_H()
c = ZZ(0)
for d in [d for d in divisors(N) if d**2 <= N]:
Nd = lcm(d,N//d)
Hd = set([x%Nd for x in H])
if Nd - 1 not in Hd:
summand = euler_phi(d)*euler_phi(N//d)//(2*len(Hd))
if d**2==N:
c = c + summand
else:
c = c + 2*summand
return c
def lift2smallest_field2(a):
"""
INPUT: a is an element of a finite field GF(q) OUTPUT: the element
b of the smallest subfield F of GF(q) for which F(b)=a.
EXAMPLES::
sage: from sage.coding.code_constructions import lift2smallest_field2
sage: FF.<z> = GF(3^4,"z")
sage: a = z^40
sage: lift2smallest_field2(a)
(2, Finite Field of size 3)
sage: FF.<z> = GF(2^4,"z")
sage: a = z^15
sage: lift2smallest_field2(a)
(1, Finite Field of size 2)
.. warning::
Since coercion (the FF(b) step) has a bug in it, this
*only works* in the case when you *know* F is a prime field.
AUTHORS:
- David Joyner
"""
FF = a.parent()
q = FF.order()
if q.is_prime():
return a, FF
p = q.factor()[0][0]
k = q.factor()[0][1]
for d in divisors(k):
F = GF(p**d, "zz")
for b in F:
if FF(b) == a:
return b, F
def lift2smallest_field2(a):
"""
INPUT: a is an element of a finite field GF(q) OUTPUT: the element
b of the smallest subfield F of GF(q) for which F(b)=a.
EXAMPLES::
sage: from sage.coding.code_constructions import lift2smallest_field2
sage: FF.<z> = GF(3^4,"z")
sage: a = z^40
sage: lift2smallest_field2(a)
(2, Finite Field of size 3)
sage: FF.<z> = GF(2^4,"z")
sage: a = z^15
sage: lift2smallest_field2(a)
(1, Finite Field of size 2)
.. warning::
Since coercion (the FF(b) step) has a bug in it, this
*only works* in the case when you *know* F is a prime field.
AUTHORS:
- David Joyner
"""
FF = a.parent()
q = FF.order()
if q.is_prime():
return a,FF
p = q.factor()[0][0]
k = q.factor()[0][1]
for d in divisors(k):
F = GF(p**d,"zz")
for b in F:
if FF(b) == a:
return b, F
def _cl_term(n, R=RationalField()):
r"""
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([n // d]).plethysm(_cl_term(d, R))
for d in divisors(n)[:-1]))
return res
def _coset_reduction_data_first_coord(G):
"""
Compute data used for determining the canonical coset
representative of an element of SL_2(Z) modulo G. This
function specifically returns data needed for the first part
of the reduction step (the first coordinate).
INPUT:
G -- a congruence subgroup Gamma_0(N), Gamma_1(N), or Gamma_H(N).
OUTPUT:
A list v such that
v[u] = (min(u*h: h in H),
gcd(u,N) ,
an h such that h*u = min(u*h: h in H)).
EXAMPLES::
sage: G = Gamma0(12)
sage: sage.modular.arithgroup.congroup_gammaH.GammaH_class._coset_reduction_data_first_coord(G)
[(0, 12, 0), (1, 1, 1), (2, 2, 1), (3, 3, 1), (4, 4, 1), (1, 1, 5), (6, 6, 1),
(1, 1, 7), (4, 4, 5), (3, 3, 7), (2, 2, 5), (1, 1, 11)]
"""
H = [ int(x) for x in G._list_of_elements_in_H() ]
N = int(G.level())
# Get some useful fast functions for inverse and gcd
inverse_mod = get_inverse_mod(N) # optimal inverse function
gcd = get_gcd(N) # optimal gcd function
# We will be filling this list in below.
reduct_data = [0] * N
# We can fill in 0 and all elements of H immediately
reduct_data[0] = (0,N,0)
for u in H:
reduct_data[u] = (1, 1, inverse_mod(u, N))
# Make a table of the reduction of H (mod N/d), one for each
# divisor d.
repr_H_mod_N_over_d = {}
for d in divisors(N):
# We special-case N == d because in this case,
# 1 % N_over_d is 0
if N == d:
repr_H_mod_N_over_d[d] = [1]
break
N_over_d = N//d
# For each element of H, we look at its image mod
# N_over_d. If we haven't yet seen it, add it on to
# the end of z.
w = [0] * N_over_d
z = [1]
for x in H:
val = x%N_over_d
if not w[val]:
w[val] = 1
z.append(x)
repr_H_mod_N_over_d[d] = z
# Compute the rest of the tuples. The values left to process
# are those where reduct_data has a 0. Note that several of
# these values are processed on each loop below, so re-index
# each time.
while True:
try:
u = reduct_data.index(0)
except ValueError:
break
d = gcd(u, N)
for x in repr_H_mod_N_over_d[d]:
reduct_data[(u*x)%N] = (u, d, inverse_mod(x,N))
return reduct_data
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 = builtins.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 __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 basis(self, reduce=True):
r"""
Produce a basis for the free abelian group of eta-products of level
N (under multiplication), attempting to find basis vectors of the
smallest possible degree.
INPUT:
- ``reduce`` - a boolean (default True) indicating
whether or not to apply LLL-reduction to the calculated basis
EXAMPLE::
sage: EtaGroup(5).basis()
[Eta product of level 5 : (eta_1)^6 (eta_5)^-6]
sage: EtaGroup(12).basis()
[Eta product of level 12 : (eta_1)^2 (eta_2)^1 (eta_3)^2 (eta_4)^-1 (eta_6)^-7 (eta_12)^3,
Eta product of level 12 : (eta_1)^-4 (eta_2)^2 (eta_3)^4 (eta_6)^-2,
Eta product of level 12 : (eta_1)^-1 (eta_2)^3 (eta_3)^3 (eta_4)^-2 (eta_6)^-9 (eta_12)^6,
Eta product of level 12 : (eta_1)^1 (eta_2)^-1 (eta_3)^-3 (eta_4)^-2 (eta_6)^7 (eta_12)^-2,
Eta product of level 12 : (eta_1)^-6 (eta_2)^9 (eta_3)^2 (eta_4)^-3 (eta_6)^-3 (eta_12)^1]
sage: EtaGroup(12).basis(reduce=False) # much bigger coefficients
[Eta product of level 12 : (eta_2)^24 (eta_12)^-24,
Eta product of level 12 : (eta_1)^-336 (eta_2)^576 (eta_3)^696 (eta_4)^-216 (eta_6)^-576 (eta_12)^-144,
Eta product of level 12 : (eta_1)^-8 (eta_2)^-2 (eta_6)^2 (eta_12)^8,
Eta product of level 12 : (eta_1)^1 (eta_2)^9 (eta_3)^13 (eta_4)^-4 (eta_6)^-15 (eta_12)^-4,
Eta product of level 12 : (eta_1)^15 (eta_2)^-24 (eta_3)^-29 (eta_4)^9 (eta_6)^24 (eta_12)^5]
ALGORITHM: An eta product of level `N` is uniquely
determined by the integers `r_d` for `d | N` with
`d < N`, since `\sum_{d | N} r_d = 0`. The valid
`r_d` are those that satisfy two congruences modulo 24,
and one congruence modulo 2 for every prime divisor of N. We beef
up the congruences modulo 2 to congruences modulo 24 by multiplying
by 12. To calculate the kernel of the ensuing map
`\ZZ^m \to (\ZZ/24\ZZ)^n`
we lift it arbitrarily to an integer matrix and calculate its Smith
normal form. This gives a basis for the lattice.
This lattice typically contains "large" elements, so by default we
pass it to the reduce_basis() function which performs
LLL-reduction to give a more manageable basis.
"""
N = self.level()
divs = divisors(N)[:-1]
s = len(divs)
primedivs = prime_divisors(N)
rows = []
for i in xrange(s):
# generate a row of relation matrix
row = [ Mod(divs[i], 24) - Mod(N, 24), Mod(N/divs[i], 24) - Mod(1, 24)]
for p in primedivs:
row.append( Mod(12*(N/divs[i]).valuation(p), 24))
rows.append(row)
M = matrix(IntegerModRing(24), rows)
Mlift = M.change_ring(ZZ)
# now we compute elementary factors of Mlift
S,U,V = Mlift.smith_form()
good_vects = []
for i in xrange(U.nrows()):
vect = U.row(i)
nf = (i < S.ncols() and S[i,i]) or 0
good_vects.append((vect * 24/gcd(nf, 24)).list())
for v in good_vects:
v.append(-sum([r for r in v]))
dicts = []
for v in good_vects:
dicts.append({})
for i in xrange(s):
dicts[-1][divs[i]] = v[i]
dicts[-1][N] = v[-1]
if reduce:
return self.reduce_basis([ self(d) for d in dicts])
else:
return [self(d) for d in dicts]
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 Möbius 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 is None:
return GammaH_class.dimension_new_cusp_forms(self, k, p)
N = self.level()
eps = DirichletGroup(N, eps.base_ring())(eps)
if eps.is_trivial():
from .all import Gamma0
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 __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 basis(self, reduce=True):
r"""
Produce a basis for the free abelian group of eta-products of level
N (under multiplication), attempting to find basis vectors of the
smallest possible degree.
INPUT:
- ``reduce`` - a boolean (default True) indicating
whether or not to apply LLL-reduction to the calculated basis
EXAMPLES::
sage: EtaGroup(5).basis()
[Eta product of level 5 : (eta_1)^6 (eta_5)^-6]
sage: EtaGroup(12).basis()
[Eta product of level 12 : (eta_1)^2 (eta_2)^1 (eta_3)^2 (eta_4)^-1 (eta_6)^-7 (eta_12)^3,
Eta product of level 12 : (eta_1)^-4 (eta_2)^2 (eta_3)^4 (eta_6)^-2,
Eta product of level 12 : (eta_1)^-1 (eta_2)^3 (eta_3)^3 (eta_4)^-2 (eta_6)^-9 (eta_12)^6,
Eta product of level 12 : (eta_1)^1 (eta_2)^-1 (eta_3)^-3 (eta_4)^-2 (eta_6)^7 (eta_12)^-2,
Eta product of level 12 : (eta_1)^-6 (eta_2)^9 (eta_3)^2 (eta_4)^-3 (eta_6)^-3 (eta_12)^1]
sage: EtaGroup(12).basis(reduce=False) # much bigger coefficients
[Eta product of level 12 : (eta_2)^24 (eta_12)^-24,
Eta product of level 12 : (eta_1)^-336 (eta_2)^576 (eta_3)^696 (eta_4)^-216 (eta_6)^-576 (eta_12)^-144,
Eta product of level 12 : (eta_1)^-8 (eta_2)^-2 (eta_6)^2 (eta_12)^8,
Eta product of level 12 : (eta_1)^1 (eta_2)^9 (eta_3)^13 (eta_4)^-4 (eta_6)^-15 (eta_12)^-4,
Eta product of level 12 : (eta_1)^15 (eta_2)^-24 (eta_3)^-29 (eta_4)^9 (eta_6)^24 (eta_12)^5]
ALGORITHM: An eta product of level `N` is uniquely
determined by the integers `r_d` for `d | N` with
`d < N`, since `\sum_{d | N} r_d = 0`. The valid
`r_d` are those that satisfy two congruences modulo 24,
and one congruence modulo 2 for every prime divisor of N. We beef
up the congruences modulo 2 to congruences modulo 24 by multiplying
by 12. To calculate the kernel of the ensuing map
`\ZZ^m \to (\ZZ/24\ZZ)^n`
we lift it arbitrarily to an integer matrix and calculate its Smith
normal form. This gives a basis for the lattice.
This lattice typically contains "large" elements, so by default we
pass it to the reduce_basis() function which performs
LLL-reduction to give a more manageable basis.
"""
from six.moves import range
N = self.level()
divs = divisors(N)[:-1]
s = len(divs)
primedivs = prime_divisors(N)
rows = []
for i in range(s):
# generate a row of relation matrix
row = [
Mod(divs[i], 24) - Mod(N, 24),
Mod(N / divs[i], 24) - Mod(1, 24)
]
for p in primedivs:
row.append(Mod(12 * (N / divs[i]).valuation(p), 24))
rows.append(row)
M = matrix(IntegerModRing(24), rows)
Mlift = M.change_ring(ZZ)
# now we compute elementary factors of Mlift
S, U, V = Mlift.smith_form()
good_vects = []
for i in range(U.nrows()):
vect = U.row(i)
nf = (i < S.ncols() and S[i, i]) or 0
good_vects.append((vect * 24 / gcd(nf, 24)).list())
for v in good_vects:
v.append(-sum([r for r in v]))
dicts = []
for v in good_vects:
dicts.append({})
for i in range(s):
dicts[-1][divs[i]] = v[i]
dicts[-1][N] = v[-1]
if reduce:
return self.reduce_basis([self(d) for d in dicts])
else:
return [self(d) for d in dicts]
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 _coset_reduction_data_first_coord(G):
"""
Compute data used for determining the canonical coset
representative of an element of SL_2(Z) modulo G. This
function specifically returns data needed for the first part
of the reduction step (the first coordinate).
INPUT:
G -- a congruence subgroup Gamma_0(N), Gamma_1(N), or Gamma_H(N).
OUTPUT:
A list v such that
v[u] = (min(u*h: h in H),
gcd(u,N) ,
an h such that h*u = min(u*h: h in H)).
EXAMPLES::
sage: G = Gamma0(12)
sage: sage.modular.arithgroup.congroup_gammaH.GammaH_class._coset_reduction_data_first_coord(G)
[(0, 12, 0), (1, 1, 1), (2, 2, 1), (3, 3, 1), (4, 4, 1), (1, 1, 5), (6, 6, 1),
(1, 1, 7), (4, 4, 5), (3, 3, 7), (2, 2, 5), (1, 1, 11)]
"""
H = [int(x) for x in G._list_of_elements_in_H()]
N = int(G.level())
# Get some useful fast functions for inverse and gcd
inverse_mod = get_inverse_mod(N) # optimal inverse function
gcd = get_gcd(N) # optimal gcd function
# We will be filling this list in below.
reduct_data = [0] * N
# We can fill in 0 and all elements of H immediately
reduct_data[0] = (0, N, 0)
for u in H:
reduct_data[u] = (1, 1, inverse_mod(u, N))
# Make a table of the reduction of H (mod N/d), one for each
# divisor d.
repr_H_mod_N_over_d = {}
for d in divisors(N):
# We special-case N == d because in this case,
# 1 % N_over_d is 0
if N == d:
repr_H_mod_N_over_d[d] = [1]
break
N_over_d = N // d
# For each element of H, we look at its image mod
# N_over_d. If we haven't yet seen it, add it on to
# the end of z.
w = [0] * N_over_d
z = [1]
for x in H:
val = x % N_over_d
if not w[val]:
w[val] = 1
z.append(x)
repr_H_mod_N_over_d[d] = z
# Compute the rest of the tuples. The values left to process
# are those where reduct_data has a 0. Note that several of
# these values are processed on each loop below, so re-index
# each time.
while True:
try:
u = reduct_data.index(0)
except ValueError:
break
d = gcd(u, N)
for x in repr_H_mod_N_over_d[d]:
reduct_data[(u * x) % N] = (u, d, inverse_mod(x, N))
return reduct_data
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 Möbius 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 is None:
return GammaH_class.dimension_new_cusp_forms(self, k, p)
N = self.level()
eps = DirichletGroup(N, eps.base_ring())(eps)
if eps.is_trivial():
from all import Gamma0
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 is_cycle(seq):
length = len(seq)
for n in divisors(length):
if n < length and is_cycle_of_length(seq, n):
return True
return False
def product_space(chi, k, weights=False, base_ring=None, verbose=False):
r"""
Computes all eisenstein series, and products of pairs of eisenstein series
of lower weight, lying in the space of modular forms of weight $k$ and
nebentypus $\chi$.
INPUT:
- chi - Dirichlet character, the nebentypus of the target space
- k - an integer, the weight of the target space
OUTPUT:
- a matrix of coefficients of q-expansions, which are the products of
Eisenstein series in M_k(chi).
WARNING: It is only for principal chi that we know that the resulting
space is the whole space of modular forms.
"""
if weights == False:
weights = srange(1, k / 2 + 1)
weight_dict = {}
weight_dict[-1] = [w for w in weights if w % 2] # Odd weights
weight_dict[1] = [w for w in weights if not w % 2] # Even weights
try:
N = chi.modulus()
except AttributeError:
if chi.parent() == ZZ:
N = chi
chi = DirichletGroup(N)[0]
Id = DirichletGroup(1)[0]
if chi(-1) != (-1)**k:
raise ValueError('chi(-1)!=(-1)^k')
sturm = ModularForms(N, k).sturm_bound() + 1
if N > 1:
target_dim = dimension_modular_forms(chi, k)
else:
target_dim = dimension_modular_forms(1, k)
D = DirichletGroup(N)
# product_space should ideally be called over number fields. Over complex
# numbers the exact linear algebra solutions might not exist.
if base_ring == None:
base_ring = CyclotomicField(euler_phi(N))
Q = PowerSeriesRing(base_ring, 'q')
q = Q.gen()
d = len(D)
prim_chars = [phi.primitive_character() for phi in D]
divs = divisors(N)
products = Matrix(base_ring, [])
indexlist = []
rank = 0
if verbose:
print(D)
print('Sturm bound', sturm)
#TODO: target_dim needs refinment in the case of weight 2.
print('Target dimension', target_dim)
for i in srange(0, d): # First character
phi = prim_chars[i]
M1 = phi.conductor()
for j in srange(0, d): # Second character
psi = prim_chars[j]
M2 = psi.conductor()
if not M1 * M2 in divs:
continue
parity = psi(-1) * phi(-1)
for t1 in divs:
if not M1 * M2 * t1 in divs:
continue
#TODO: THE NEXT CONDITION NEEDS TO BE CORRECTED. THIS IS JUST A TEST
if phi.bar() == psi and not (
k == 2): #and i==0 and j==0 and t1==1):
E = eisenstein_series_at_inf(phi, psi, k, sturm, t1,
base_ring).padded_list()
try:
products.T.solve_right(vector(base_ring, E))
except ValueError:
products = Matrix(products.rows() + [E])
indexlist.append([k, i, j, t1])
rank += 1
if verbose:
print('Added ', [k, i, j, t1])
print('Rank is now', rank)
if rank == target_dim:
return products, indexlist
for t in divs:
if not M1 * M2 * t1 * t in divs:
continue
for t2 in divs:
if not M1 * M2 * t1 * t2 * t in divs:
continue
for l in weight_dict[parity]:
if l == 1 and phi.is_odd():
continue
if i == 0 and j == 0 and (l == 2 or l == k - 2):
continue
#TODO: THE NEXT CONDITION NEEDS TO BE REMOVED. THIS IS JUST A TEST
if l == 2 or l == k - 2:
continue
E1 = eisenstein_series_at_inf(
phi, psi, l, sturm, t1 * t, base_ring)
E2 = eisenstein_series_at_inf(
phi**(-1), psi**(-1), k - l, sturm, t2 * t,
base_ring)
#If chi is non-principal this needs to be changed to be something like chi*phi^(-1) instead of phi^(-1)
E = (E1 * E2 + O(q**sturm)).padded_list()
try:
products.T.solve_right(vector(base_ring, E))
except ValueError:
products = Matrix(products.rows() + [E])
indexlist.append([l, k - l, i, j, t1, t2, t])
rank += 1
if verbose:
print('Added ',
[l, k - l, i, j, t1, t2, t])
print('Rank', rank)
if rank == target_dim:
return products, indexlist
return products, indexlist
def e_phipsi(phi,
psi,
k,
t=1,
prec=5,
mat=Matrix([[1, 0], [0, 1]]),
base_ring=None):
r"""
Computes the Eisenstein series attached to the characters psi, phi as
defined on p129 of Diamond--Shurman hit by mat \in \SL_2(\Z)
INPUT:
- psi, a primitive Dirichlet character
- phi, a primitive Dirichlet character
- k, int -- the weight
- t, int -- the shift
- prec, the desired absolute precision, can be fractional. The expansion will be up to O(q_w^(floor(w*prec))), where w is the width of the cusp.
- mat, a matrix - typically taking $i\infty$ to some other cusp
- base_ring, a ring - the ring in which the modular forms are defined
OUTPUT:
- an instance of QExpansion
"""
chi2 = phi
chi1 = psi
try:
assert (QQ(chi1(-1)) * QQ(chi2(-1)) == (-1)**k)
except AssertionError:
print("Parity of characters must match parity of weight")
return None
N1 = chi1.level()
N2 = chi2.level()
N = t * N1 * N2
mat2, Tn = find_correct_matrix(mat, N)
#By construction gamma = mat2 * Tn * mat**(-1) is in Gamma0(N) so if E is our Eisenstein series we can evaluate E|mat = chi(gamma) * E|mat2*Tn.
#Since E|mat2 has a Fourier expansion in qN, the matrix Tn acts as a a twist. The value c_gamma = chi(gamma) is calculated below.
#The point of swapping mat with mat2 is that mat2 satisfies C|N, C>0, (A,N)=1 and N|B and our formulas for the coefficients require this condition.
A, B, C, D = mat2.list()
gamma = mat2 * Tn * mat**(-1)
if base_ring == None:
Nbig = lcm(N, euler_phi(N1 * N2))
base_ring = CyclotomicField(Nbig, 'zeta' + str(Nbig))
zetaNbig = base_ring.gen()
zetaN = zetaNbig**(Nbig / N)
else:
zetaN = base_ring.zeta(N)
g = gcd(t, C)
g1 = gcd(N1 * g, C)
g2 = gcd(N2 * g, C)
#Resulting Eisenstein series will have Fourier expansion in q_N**(g1g2)=q_(N/gcd(N,g1g2))**(g1g2/gcd(N,g1g2))
Q = PowerSeriesRing(base_ring, 'q' + str(N / gcd(N, g1 * g2)))
qN = Q.gen()
zeta_Cg = zetaN**(N / (C / g))
zeta_tmp = zeta_Cg**(inverse_mod(-A * ZZ(t / g), C / g))
#Calculating a few values that will be used repeatedly
chi1bar_vals = [base_ring(chi1.bar()(i)) for i in range(N1)]
cp_list1 = [i for i in range(N1) if gcd(i, N1) == 1]
chi2bar_vals = [base_ring(chi2.bar()(i)) for i in range(N2)]
cp_list2 = [i for i in range(N2) if gcd(i, N2) == 1]
#Computation of the Fourier coefficients
ser = O(qN**floor(prec * N / gcd(N, g1 * g2)))
for n in range(1, ceil(prec * N / QQ(g1 * g2)) + 1):
f = 0
for m in divisors(n) + list(map(lambda x: -x, divisors(n))):
a = 0
for r1 in cp_list1:
b = 0
if ((C / g1) * r1 - QQ(n) / QQ(m)) % ((N1 * g) / g1) == 0:
for r2 in cp_list2:
if ((C / g2) * r2 - m) % ((N2 * g) / g2) == 0:
b += chi2bar_vals[r2] * zeta_tmp**(
(n / m - (C / g1) * r1) / ((N1 * g) / g1) *
(m - (C / g2) * r2) / ((N2 * g) / g2))
a += chi1bar_vals[r1] * b
a *= sign(m) * m**(k - 1)
f += a
f *= zetaN**(inverse_mod(A, N) * (g1 * g2) / C * n)
#The additional factor zetaN**(n*Tn[0][1]) comes from the twist by Tn
ser += zetaN**(n * g1 * g2 * Tn[0][1]) * f * qN**(n * (
(g1 * g2) / gcd(N, g1 * g2)))
#zk(chi1, chi2, c)
gauss1 = base_ring(gauss_sum_corr(chi1.bar()))
gauss2 = base_ring(gauss_sum_corr(chi2.bar()))
zk = 2 * (N2 * t / QQ(g2))**(k - 1) * (t / g) * gauss1 * gauss2
#The following is a temporary fix for a bug in sage
if base_ring == CC:
G = DirichletGroup(N1 * N2, CC)
G[0] #Otherwise chi1.bar().extend(N1*N2).base_extend(CC) or chi2.bar().extend(N1*N2).base_extend(CC) will produce an error
#Constant term
#c_gamma comes from replacing mat with mat2.
c_gamma = chi1bar_vals[gamma[1][1] % N1] * chi2bar_vals[gamma[1][1] % N2]
if N1.divides(C) and ZZ(C / N1).divides(t) and gcd(t / (C / N1), N1) == 1:
ser += (-1)**(k - 1) * gauss2 / QQ(
N2 * (g2 / g)**(k - 1)) * chi1bar_vals[(-A * t / g) % N1] * Sk(
chi1.bar().extend(N1 * N2).base_extend(base_ring) *
chi2.extend(N1 * N2).base_extend(base_ring), k)
elif k == 1 and N2.divides(C) and ZZ(C / N2).divides(t) and gcd(
t / (C / N2), N2) == 1:
ser += gauss1 / QQ(N1) * chi2bar_vals[(-A * t / g) % N2] * Sk(
chi1.extend(N1 * N2).base_extend(base_ring) *
chi2.bar().extend(N1 * N2).base_extend(base_ring), k)
return QExpansion(
N, k,
2 / zk * c_gamma * ser + O(qN**floor(prec * N / gcd(N, g1 * g2))),
N / gcd(N, g1 * g2))
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(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 = cputime()
for e in G:
m = Integer(e.conductor())
if m in C:
C[m].append(e)
else:
C[m] = [e]
verbose("Enumeration with conductors.", t0)
params = []
for L in divisors(N):
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 regular_symmetric_hadamard_matrix_with_constant_diagonal(n,e,existence=False):
r"""
Return a Regular Symmetric Hadamard Matrix with Constant Diagonal.
A Hadamard matrix is said to be *regular* if its rows all sum to the same
value.
For `\epsilon\in\{-1,+1\}`, we say that `M` is a `(n,\epsilon)-RSHCD` if
`M` is a regular symmetric Hadamard matrix with constant diagonal
`\delta\in\{-1,+1\}` and row sums all equal to `\delta \epsilon
\sqrt(n)`. For more information, see [HX10]_ or 10.5.1 in
[BH12]_. For the case `n=324`, see :func:`RSHCD_324` and [CP16]_.
INPUT:
- ``n`` (integer) -- side of the matrix
- ``e`` -- one of `-1` or `+1`, equal to the value of `\epsilon`
EXAMPLES::
sage: from sage.combinat.matrices.hadamard_matrix import regular_symmetric_hadamard_matrix_with_constant_diagonal
sage: regular_symmetric_hadamard_matrix_with_constant_diagonal(4,1)
[ 1 1 1 -1]
[ 1 1 -1 1]
[ 1 -1 1 1]
[-1 1 1 1]
sage: regular_symmetric_hadamard_matrix_with_constant_diagonal(4,-1)
[ 1 -1 -1 -1]
[-1 1 -1 -1]
[-1 -1 1 -1]
[-1 -1 -1 1]
Other hardcoded values::
sage: for n,e in [(36,1),(36,-1),(100,1),(100,-1),(196, 1)]:
....: print regular_symmetric_hadamard_matrix_with_constant_diagonal(n,e)
36 x 36 dense matrix over Integer Ring
36 x 36 dense matrix over Integer Ring
100 x 100 dense matrix over Integer Ring
100 x 100 dense matrix over Integer Ring
196 x 196 dense matrix over Integer Ring
sage: for n,e in [(324,1),(324,-1)]: # not tested - long time, tested in RSHCD_324
....: print regular_symmetric_hadamard_matrix_with_constant_diagonal(n,e) # not tested - long time
324 x 324 dense matrix over Integer Ring
324 x 324 dense matrix over Integer Ring
From two close prime powers::
sage: print regular_symmetric_hadamard_matrix_with_constant_diagonal(64,-1)
64 x 64 dense matrix over Integer Ring
Recursive construction::
sage: print regular_symmetric_hadamard_matrix_with_constant_diagonal(144,-1)
144 x 144 dense matrix over Integer Ring
REFERENCE:
.. [BH12] \A. Brouwer and W. Haemers,
Spectra of graphs,
Springer, 2012,
http://homepages.cwi.nl/~aeb/math/ipm/ipm.pdf
.. [HX10] \W. Haemers and Q. Xiang,
Strongly regular graphs with parameters `(4m^4,2m^4+m^2,m^4+m^2,m^4+m^2)` exist for all `m>1`,
European Journal of Combinatorics,
Volume 31, Issue 6, August 2010, Pages 1553-1559,
http://dx.doi.org/10.1016/j.ejc.2009.07.009.
"""
if existence and (n,e) in _rshcd_cache:
return _rshcd_cache[n,e]
from sage.graphs.strongly_regular_db import strongly_regular_graph
def true():
_rshcd_cache[n,e] = True
return True
M = None
if abs(e) != 1:
raise ValueError
if n<0:
if existence:
return False
raise ValueError
elif n == 4:
if existence:
return true()
if e == 1:
M = J(4)-2*matrix(4,[[int(i+j == 3) for i in range(4)] for j in range(4)])
else:
M = -J(4)+2*I(4)
elif n == 36:
if existence:
return true()
if e == 1:
M = strongly_regular_graph(36, 15, 6, 6).adjacency_matrix()
M = J(36) - 2*M
else:
M = strongly_regular_graph(36,14,4,6).adjacency_matrix()
M = -J(36) + 2*M + 2*I(36)
elif n == 100:
if existence:
return true()
if e == -1:
M = strongly_regular_graph(100,44,18,20).adjacency_matrix()
M = 2*M - J(100) + 2*I(100)
else:
M = strongly_regular_graph(100,45,20,20).adjacency_matrix()
M = J(100) - 2*M
elif n == 196 and e == 1:
if existence:
return true()
M = strongly_regular_graph(196,91,42,42).adjacency_matrix()
M = J(196) - 2*M
elif n == 324:
if existence:
return true()
M = RSHCD_324(e)
elif ( e == 1 and
n%16 == 0 and
is_square(n) and
is_prime_power(sqrt(n)-1) and
is_prime_power(sqrt(n)+1)):
if existence:
return true()
M = -rshcd_from_close_prime_powers(int(sqrt(n)))
# Recursive construction: the kronecker product of two RSHCD is a RSHCD
else:
from itertools import product
for n1,e1 in product(divisors(n)[1:-1],[-1,1]):
e2 = e1*e
n2 = n//n1
if (regular_symmetric_hadamard_matrix_with_constant_diagonal(n1,e1,existence=True) and
regular_symmetric_hadamard_matrix_with_constant_diagonal(n2,e2,existence=True)):
if existence:
return true()
M1 = regular_symmetric_hadamard_matrix_with_constant_diagonal(n1,e1)
M2 = regular_symmetric_hadamard_matrix_with_constant_diagonal(n2,e2)
M = M1.tensor_product(M2)
break
if M is None:
from sage.misc.unknown import Unknown
_rshcd_cache[n,e] = Unknown
if existence:
return Unknown
raise ValueError("I do not know how to build a {}-RSHCD".format((n,e)))
assert M*M.transpose() == n*I(n)
assert set(map(sum,M)) == {e*sqrt(n)}
return M
def enumerate_totallyreal_fields_all(n,
B,
verbose=0,
return_seqs=False,
return_pari_objects=True):
r"""
Enumerates *all* totally real fields of degree ``n`` with discriminant
at most ``B``, primitive or otherwise.
INPUT:
- ``n`` -- integer, the degree
- ``B`` -- integer, the discriminant bound
- ``verbose`` -- boolean or nonnegative integer or string (default: 0)
give a verbose description of the computations being performed. If
``verbose`` is set to ``2`` or more then it outputs some extra
information. If ``verbose`` is a string then it outputs to a file
specified by ``verbose``
- ``return_seqs`` -- (boolean, default False) If ``True``, then return
the polynomials as sequences (for easier exporting to a file). This
also returns a list of four numbers, as explained in the OUTPUT
section below.
- ``return_pari_objects`` -- (boolean, default: True) if both
``return_seqs`` and ``return_pari_objects`` are ``False`` then it
returns the elements as Sage objects; otherwise it returns pari
objects.
EXAMPLES::
sage: enumerate_totallyreal_fields_all(4, 2000)
[[725, x^4 - x^3 - 3*x^2 + x + 1],
[1125, x^4 - x^3 - 4*x^2 + 4*x + 1],
[1600, x^4 - 6*x^2 + 4],
[1957, x^4 - 4*x^2 - x + 1],
[2000, x^4 - 5*x^2 + 5]]
sage: enumerate_totallyreal_fields_all(1, 10)
[[1, x - 1]]
TESTS:
Each of the outputs must be elements of Sage if ``return_pari_objects``
is set to ``False``::
sage: enumerate_totallyreal_fields_all(2, 10)
[[5, x^2 - x - 1], [8, x^2 - 2]]
sage: type(enumerate_totallyreal_fields_all(2, 10)[0][1])
<type 'cypari2.gen.Gen'>
sage: enumerate_totallyreal_fields_all(2, 10, return_pari_objects=False)[0][1].parent()
Univariate Polynomial Ring in x over Rational Field
In practice most of these will be found by
:func:`~sage.rings.number_field.totallyreal.enumerate_totallyreal_fields_prim`,
which is guaranteed to return all primitive fields but often returns
many non-primitive ones as well. For instance, only one of the five
fields in the example above is primitive, but
:func:`~sage.rings.number_field.totallyreal.enumerate_totallyreal_fields_prim`
finds four out of the five (the exception being `x^4 - 6x^2 + 4`).
The following was fixed in :trac:`13101`::
sage: enumerate_totallyreal_fields_all(8, 10^6) # long time (about 2 s)
[]
"""
S = []
counts = [0, 0, 0, 0]
if len(divisors(n)) > 4:
raise ValueError("Only implemented for n = p*q with p,q prime")
for d in divisors(n):
if d > 1 and d < n:
Sds = enumerate_totallyreal_fields_prim(
d, int(math.floor((1. * B)**(1. * d / n))), verbose=verbose)
for i in range(len(Sds)):
if verbose:
print("=" * 80)
print("Taking F =", Sds[i][1])
F = NumberField(ZZx(Sds[i][1]), 't')
T = enumerate_totallyreal_fields_rel(F,
n / d,
B,
verbose=verbose,
return_seqs=return_seqs)
if return_seqs:
for i in range(4):
counts[i] += T[0][i]
S += [[t[0], pari(t[1]).Polrev()] for t in T[1]]
else:
S += [[t[0], t[1]] for t in T]
for E in enumerate_totallyreal_fields_prim(
n / d,
int(
math.floor((1. * B)**(1. / d) /
(1. * Sds[i][0])**(n * 1. / d**2)))):
for EF in F.composite_fields(NumberField(ZZx(E[1]), 'u')):
if EF.degree() == n and EF.disc() <= B:
S.append(
[EF.disc(),
pari(EF.absolute_polynomial())])
S += enumerate_totallyreal_fields_prim(n, B, verbose=verbose)
S.sort(key=lambda x: (x[0], [QQ(cf) for cf in x[1].polrecip().Vec()]))
weed_fields(S)
# Output.
if verbose:
saveout = sys.stdout
if isinstance(verbose, str):
fsock = open(verbose, 'w')
sys.stdout = fsock
# Else, print to screen
print("=" * 80)
print("Polynomials tested: {}".format(counts[0]))
print("Polynomials with discriminant with large enough square"
" divisor: {}".format(counts[1]))
print("Irreducible polynomials: {}".format(counts[2]))
print("Polynomials with nfdisc <= B: {}".format(counts[3]))
for i in range(len(S)):
print(S[i])
if isinstance(verbose, str):
fsock.close()
sys.stdout = saveout
# Make sure to return elements that belong to Sage
if return_seqs:
return [[ZZ(_) for _ in counts],
[[ZZ(s[0]), [QQ(_) for _ in s[1].polrecip().Vec()]]
for s in S]]
elif return_pari_objects:
return S
else:
Px = PolynomialRing(QQ, 'x')
return [[ZZ(s[0]), Px([QQ(_) for _ in s[1].list()])] for s in S]
def skew_hadamard_matrix(n,existence=False, skew_normalize=True, check=True):
r"""
Tries to construct a skew Hadamard matrix
A Hadamard matrix `H` is called skew if `H=S-I`, for `I` the identity matrix
and `-S=S^\top`. Currently constructions from Section 14.1 of [Ha83]_ and few
more exotic ones are implemented.
INPUT:
- ``n`` (integer) -- dimension of the matrix
- ``existence`` (boolean) -- whether to build the matrix or merely query if
a construction is available in Sage. When set to ``True``, the function
returns:
- ``True`` -- meaning that Sage knows how to build the matrix
- ``Unknown`` -- meaning that Sage does not know how to build the
matrix, but that the design may exist (see :mod:`sage.misc.unknown`).
- ``False`` -- meaning that the matrix does not exist.
- ``skew_normalize`` (boolean) -- whether to make the 1st row all-one, and
adjust the 1st column accordingly. Set to ``True`` by default.
- ``check`` (boolean) -- whether to check that output is correct before
returning it. As this is expected to be useless (but we are cautious
guys), you may want to disable it whenever you want speed. Set to ``True``
by default.
EXAMPLES::
sage: from sage.combinat.matrices.hadamard_matrix import skew_hadamard_matrix
sage: skew_hadamard_matrix(12).det()
2985984
sage: 12^6
2985984
sage: skew_hadamard_matrix(1)
[1]
sage: skew_hadamard_matrix(2)
[ 1 1]
[-1 1]
TESTS::
sage: skew_hadamard_matrix(10,existence=True)
False
sage: skew_hadamard_matrix(12,existence=True)
True
sage: skew_hadamard_matrix(784,existence=True)
True
sage: skew_hadamard_matrix(10)
Traceback (most recent call last):
...
ValueError: A skew Hadamard matrix of order 10 does not exist
sage: skew_hadamard_matrix(36)
36 x 36 dense matrix over Integer Ring...
sage: skew_hadamard_matrix(36)==skew_hadamard_matrix(36,skew_normalize=False)
False
sage: skew_hadamard_matrix(52)
52 x 52 dense matrix over Integer Ring...
sage: skew_hadamard_matrix(92)
92 x 92 dense matrix over Integer Ring...
sage: skew_hadamard_matrix(816) # long time
816 x 816 dense matrix over Integer Ring...
sage: skew_hadamard_matrix(100)
Traceback (most recent call last):
...
ValueError: A skew Hadamard matrix of order 100 is not yet implemented.
sage: skew_hadamard_matrix(100,existence=True)
Unknown
REFERENCES:
.. [Ha83] \M. Hall,
Combinatorial Theory,
2nd edition,
Wiley, 1983
"""
def true():
_skew_had_cache[n]=True
return True
M = None
if existence and n in _skew_had_cache:
return True
if not(n % 4 == 0) and (n > 2):
if existence:
return False
raise ValueError("A skew Hadamard matrix of order %s does not exist" % n)
if n == 2:
if existence:
return true()
M = matrix([[1, 1], [-1, 1]])
elif n == 1:
if existence:
return true()
M = matrix([1])
elif is_prime_power(n - 1) and ((n - 1) % 4 == 3):
if existence:
return true()
M = hadamard_matrix_paleyI(n, normalize=False)
elif n % 8 == 0:
if skew_hadamard_matrix(n//2,existence=True): # (Lemma 14.1.6 in [Ha83]_)
if existence:
return true()
H = skew_hadamard_matrix(n//2,check=False)
M = block_matrix([[H,H], [-H.T,H.T]])
else: # try Williamson construction (Lemma 14.1.5 in [Ha83]_)
for d in divisors(n)[2:-2]: # skip 1, 2, n/2, and n
n1 = n//d
if is_prime_power(d - 1) and (d % 4 == 0) and (n1 % 4 == 0)\
and skew_hadamard_matrix(n1,existence=True):
if existence:
return true()
H = skew_hadamard_matrix(n1, check=False)-I(n1)
U = matrix(ZZ, d, lambda i, j: -1 if i==j==0 else\
1 if i==j==1 or (i>1 and j-1==d-i)\
else 0)
A = block_matrix([[matrix([0]), matrix(ZZ,1,d-1,[1]*(d-1))],
[ matrix(ZZ,d-1,1,[-1]*(d-1)),
_helper_payley_matrix(d-1,zero_position=0)]])+I(d)
M = A.tensor_product(I(n1))+(U*A).tensor_product(H)
break
if M is None: # try Williamson-Goethals-Seidel construction
if GS_skew_hadamard_smallcases(n, existence=True):
if existence:
return true()
M = GS_skew_hadamard_smallcases(n)
else:
if existence:
return Unknown
raise ValueError("A skew Hadamard matrix of order %s is not yet implemented." % n)
if skew_normalize:
dd = diagonal_matrix(M[0])
M = dd*M*dd
if check:
assert is_hadamard_matrix(M, normalized=False, skew=True)
if skew_normalize:
from sage.modules.free_module_element import vector
assert M[0]==vector([1]*n)
_skew_had_cache[n]=True
return M
def enumerate_totallyreal_fields_all(n, B, verbose=0, return_seqs=False,
return_pari_objects=True):
r"""
Enumerates *all* totally real fields of degree ``n`` with discriminant
at most ``B``, primitive or otherwise.
INPUT:
- ``n`` -- integer, the degree
- ``B`` -- integer, the discriminant bound
- ``verbose`` -- boolean or nonnegative integer or string (default: 0)
give a verbose description of the computations being performed. If
``verbose`` is set to ``2`` or more then it outputs some extra
information. If ``verbose`` is a string then it outputs to a file
specified by ``verbose``
- ``return_seqs`` -- (boolean, default False) If ``True``, then return
the polynomials as sequences (for easier exporting to a file). This
also returns a list of four numbers, as explained in the OUTPUT
section below.
- ``return_pari_objects`` -- (boolean, default: True) if both
``return_seqs`` and ``return_pari_objects`` are ``False`` then it
returns the elements as Sage objects; otherwise it returns pari
objects.
EXAMPLES::
sage: enumerate_totallyreal_fields_all(4, 2000)
[[725, x^4 - x^3 - 3*x^2 + x + 1],
[1125, x^4 - x^3 - 4*x^2 + 4*x + 1],
[1600, x^4 - 6*x^2 + 4],
[1957, x^4 - 4*x^2 - x + 1],
[2000, x^4 - 5*x^2 + 5]]
sage: enumerate_totallyreal_fields_all(1, 10)
[[1, x - 1]]
TESTS:
Each of the outputs must be elements of Sage if ``return_pari_objects``
is set to ``False``::
sage: enumerate_totallyreal_fields_all(2, 10)
[[5, x^2 - x - 1], [8, x^2 - 2]]
sage: enumerate_totallyreal_fields_all(2, 10)[0][1].parent()
Interface to the PARI C library
sage: enumerate_totallyreal_fields_all(2, 10, return_pari_objects=False)[0][1].parent()
Univariate Polynomial Ring in x over Rational Field
In practice most of these will be found by
:func:`~sage.rings.number_field.totallyreal.enumerate_totallyreal_fields_prim`,
which is guaranteed to return all primitive fields but often returns
many non-primitive ones as well. For instance, only one of the five
fields in the example above is primitive, but
:func:`~sage.rings.number_field.totallyreal.enumerate_totallyreal_fields_prim`
finds four out of the five (the exception being `x^4 - 6x^2 + 4`).
The following was fixed in :trac:`13101`::
sage: enumerate_totallyreal_fields_all(8, 10^6) # long time (about 2 s)
[]
"""
S = []
counts = [0,0,0,0]
if len(divisors(n)) > 4:
raise ValueError("Only implemented for n = p*q with p,q prime")
for d in divisors(n):
if d > 1 and d < n:
Sds = enumerate_totallyreal_fields_prim(d, int(math.floor((1.*B)**(1.*d/n))), verbose=verbose)
for i in range(len(Sds)):
if verbose:
print "="*80
print "Taking F =", Sds[i][1]
F = NumberField(ZZx(Sds[i][1]), 't')
T = enumerate_totallyreal_fields_rel(F, n/d, B, verbose=verbose, return_seqs=return_seqs)
if return_seqs:
for i in range(4):
counts[i] += T[0][i]
S += [[t[0],pari(t[1]).Polrev()] for t in T[1]]
else:
S += [[t[0],t[1]] for t in T]
j = i+1
for E in enumerate_totallyreal_fields_prim(n/d, int(math.floor((1.*B)**(1./d)/(1.*Sds[i][0])**(n*1./d**2)))):
for EF in F.composite_fields(NumberField(ZZx(E[1]), 'u')):
if EF.degree() == n and EF.disc() <= B:
S.append([EF.disc(), pari(EF.absolute_polynomial())])
S += enumerate_totallyreal_fields_prim(n, B, verbose=verbose)
S.sort(cmp=lambda x, y: cmp(x[0], y[0]) or cmp(x[1], y[1]))
weed_fields(S)
# Output.
if verbose:
saveout = sys.stdout
if isinstance(verbose, str):
fsock = open(verbose, 'w')
sys.stdout = fsock
# Else, print to screen
print "="*80
print "Polynomials tested: {}".format(counts[0])
print ( "Polynomials with discriminant with large enough square"
" divisor: {}".format(counts[1]))
print "Irreducible polynomials: {}".format(counts[2])
print "Polynomials with nfdisc <= B: {}".format(counts[3])
for i in range(len(S)):
print S[i]
if isinstance(verbose, str):
fsock.close()
sys.stdout = saveout
# Make sure to return elements that belong to Sage
if return_seqs:
return [[ZZ(_) for _ in counts],
[[ZZ(s[0]), [QQ(_) for _ in s[1].polrecip().Vec()]] for s in S]]
elif return_pari_objects:
return S
else:
Px = PolynomialRing(QQ, 'x')
return [[ZZ(s[0]), Px([QQ(_) for _ in s[1].list()])]
for s in S]
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.
C = {}
t0 = cputime()
for e in G:
m = Integer(e.conductor())
if m in C:
C[m].append(e)
else:
C[m] = [e]
verbose("Enumeration with conductors.", t0)
params = []
for L in divisors(N):
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 skew_hadamard_matrix(n, existence=False, skew_normalize=True, check=True):
r"""
Tries to construct a skew Hadamard matrix
A Hadamard matrix `H` is called skew if `H=S-I`, for `I` the identity matrix
and `-S=S^\top`. Currently constructions from Section 14.1 of [Ha83]_ and few
more exotic ones are implemented.
INPUT:
- ``n`` (integer) -- dimension of the matrix
- ``existence`` (boolean) -- whether to build the matrix or merely query if
a construction is available in Sage. When set to ``True``, the function
returns:
- ``True`` -- meaning that Sage knows how to build the matrix
- ``Unknown`` -- meaning that Sage does not know how to build the
matrix, but that the design may exist (see :mod:`sage.misc.unknown`).
- ``False`` -- meaning that the matrix does not exist.
- ``skew_normalize`` (boolean) -- whether to make the 1st row all-one, and
adjust the 1st column accordingly. Set to ``True`` by default.
- ``check`` (boolean) -- whether to check that output is correct before
returning it. As this is expected to be useless (but we are cautious
guys), you may want to disable it whenever you want speed. Set to ``True``
by default.
EXAMPLES::
sage: from sage.combinat.matrices.hadamard_matrix import skew_hadamard_matrix
sage: skew_hadamard_matrix(12).det()
2985984
sage: 12^6
2985984
sage: skew_hadamard_matrix(1)
[1]
sage: skew_hadamard_matrix(2)
[ 1 1]
[-1 1]
TESTS::
sage: skew_hadamard_matrix(10,existence=True)
False
sage: skew_hadamard_matrix(12,existence=True)
True
sage: skew_hadamard_matrix(784,existence=True)
True
sage: skew_hadamard_matrix(10)
Traceback (most recent call last):
...
ValueError: A skew Hadamard matrix of order 10 does not exist
sage: skew_hadamard_matrix(36)
36 x 36 dense matrix over Integer Ring...
sage: skew_hadamard_matrix(36)==skew_hadamard_matrix(36,skew_normalize=False)
False
sage: skew_hadamard_matrix(52)
52 x 52 dense matrix over Integer Ring...
sage: skew_hadamard_matrix(92)
92 x 92 dense matrix over Integer Ring...
sage: skew_hadamard_matrix(816) # long time
816 x 816 dense matrix over Integer Ring...
sage: skew_hadamard_matrix(100)
Traceback (most recent call last):
...
ValueError: A skew Hadamard matrix of order 100 is not yet implemented.
sage: skew_hadamard_matrix(100,existence=True)
Unknown
REFERENCES:
.. [Ha83] \M. Hall,
Combinatorial Theory,
2nd edition,
Wiley, 1983
"""
def true():
_skew_had_cache[n] = True
return True
M = None
if existence and n in _skew_had_cache:
return True
if not (n % 4 == 0) and (n > 2):
if existence:
return False
raise ValueError("A skew Hadamard matrix of order %s does not exist" %
n)
if n == 2:
if existence:
return true()
M = matrix([[1, 1], [-1, 1]])
elif n == 1:
if existence:
return true()
M = matrix([1])
elif is_prime_power(n - 1) and ((n - 1) % 4 == 3):
if existence:
return true()
M = hadamard_matrix_paleyI(n, normalize=False)
elif n % 8 == 0:
if skew_hadamard_matrix(n // 2,
existence=True): # (Lemma 14.1.6 in [Ha83]_)
if existence:
return true()
H = skew_hadamard_matrix(n // 2, check=False)
M = block_matrix([[H, H], [-H.T, H.T]])
else: # try Williamson construction (Lemma 14.1.5 in [Ha83]_)
for d in divisors(n)[2:-2]: # skip 1, 2, n/2, and n
n1 = n // d
if is_prime_power(d - 1) and (d % 4 == 0) and (n1 % 4 == 0)\
and skew_hadamard_matrix(n1,existence=True):
if existence:
return true()
H = skew_hadamard_matrix(n1, check=False) - I(n1)
U = matrix(ZZ, d, lambda i, j: -1 if i==j==0 else\
1 if i==j==1 or (i>1 and j-1==d-i)\
else 0)
A = block_matrix(
[[matrix([0]),
matrix(ZZ, 1, d - 1, [1] * (d - 1))],
[
matrix(ZZ, d - 1, 1, [-1] * (d - 1)),
_helper_payley_matrix(d - 1, zero_position=0)
]]) + I(d)
M = A.tensor_product(I(n1)) + (U * A).tensor_product(H)
break
if M is None: # try Williamson-Goethals-Seidel construction
if GS_skew_hadamard_smallcases(n, existence=True):
if existence:
return true()
M = GS_skew_hadamard_smallcases(n)
else:
if existence:
return Unknown
raise ValueError(
"A skew Hadamard matrix of order %s is not yet implemented." %
n)
if skew_normalize:
dd = diagonal_matrix(M[0])
M = dd * M * dd
if check:
assert is_hadamard_matrix(M, normalized=False, skew=True)
if skew_normalize:
from sage.modules.free_module_element import vector
assert M[0] == vector([1] * n)
_skew_had_cache[n] = True
return M
def Gamma0_NFCusps(N):
r"""
Returns a list of inequivalent cusps for `\Gamma_0(N)`, i.e., a set of
representatives for the orbits of ``self`` on `\mathbb{P}^1(k)`.
INPUT:
- ``N`` -- an integral ideal of the number field k (the level).
OUTPUT:
A list of inequivalent number field cusps.
EXAMPLES:
::
sage: k.<a> = NumberField(x^2 + 5)
sage: N = k.ideal(3)
sage: L = Gamma0_NFCusps(N)
The cusps in the list are inequivalent:
::
sage: all([not L[i].is_Gamma0_equivalent(L[j], N) for i, j in \
mrange([len(L), len(L)]) if i<j])
True
We test that we obtain the right number of orbits:
::
sage: from sage.modular.cusps_nf import number_of_Gamma0_NFCusps
sage: len(L) == number_of_Gamma0_NFCusps(N)
True
Another example:
::
sage: k.<a> = NumberField(x^4 - x^3 -21*x^2 + 17*x + 133)
sage: N = k.ideal(5)
sage: from sage.modular.cusps_nf import number_of_Gamma0_NFCusps
sage: len(Gamma0_NFCusps(N)) == number_of_Gamma0_NFCusps(N) # long time (over 1 sec)
True
"""
# We create L a list of three lists, which are different and each a list of
# prime ideals, coprime to N, representing the ideal classes of k
L = NFCusps_ideal_reps_for_levelN(N, nlists=3)
Laux = L[1] + L[2]
Lreps = list_of_representatives(N)
Lcusps = []
k = N.number_field()
for A in L[0]:
#find B in inverse class:
if A.is_trivial():
B = k.ideal(1)
#B = k.unit_ideal() produces an error because we need fract ideal
g = 1
else:
Lbs = [P for P in Laux if (P * A).is_principal()]
B = Lbs[0]
g = (A * B).gens_reduced()[0]
#for every divisor of N we have to find cusps
from sage.arith.all import divisors
for d in divisors(N):
#find delta prime coprime to B in inverse class of d*A
#by searching in our list of auxiliary prime ideals
Lds = [
P for P in Laux
if (P * d * A).is_principal() and P.is_coprime(B)
]
deltap = Lds[0]
a = (deltap * d * A).gens_reduced()[0]
I = d + N / d
#especial case: A=B=d=<1>:
if a.is_one() and I.is_trivial():
Lcusps.append(NFCusp(k, 0, 1, lreps=Lreps))
else:
u = k.unit_group().gens()
for b in I.invertible_residues_mod(u):
#Note: if I trivial, invertible_residues_mod returns [1]
#lift b to (R/a)star
#we need the part of d which is coprime to I, call it M
M = d.prime_to_idealM_part(I)
deltAM = deltap * A * M
u = (B * deltAM).element_1_mod(I)
v = (I * B).element_1_mod(deltAM)
newb = u * b + v
#build AB-matrix:
#----> extended gcd for k.ideal(a), k.ideal(newb)
Y = k.ideal(newb).element_1_mod(k.ideal(a))
# if xa + yb = 1, cusp = y*g /a
Lcusps.append(NFCusp(k, Y * g, a, lreps=Lreps))
return Lcusps
def is_cycle(seq):
length = len(seq)
for n in divisors(length):
if n < length and is_cycle_of_length(seq, n):
return True
return False
def regular_symmetric_hadamard_matrix_with_constant_diagonal(
n, e, existence=False):
r"""
Return a Regular Symmetric Hadamard Matrix with Constant Diagonal.
A Hadamard matrix is said to be *regular* if its rows all sum to the same
value.
For `\epsilon\in\{-1,+1\}`, we say that `M` is a `(n,\epsilon)-RSHCD` if
`M` is a regular symmetric Hadamard matrix with constant diagonal
`\delta\in\{-1,+1\}` and row sums all equal to `\delta \epsilon
\sqrt(n)`. For more information, see [HX10]_ or 10.5.1 in
[BH12]_. For the case `n=324`, see :func:`RSHCD_324` and [CP16]_.
INPUT:
- ``n`` (integer) -- side of the matrix
- ``e`` -- one of `-1` or `+1`, equal to the value of `\epsilon`
EXAMPLES::
sage: from sage.combinat.matrices.hadamard_matrix import regular_symmetric_hadamard_matrix_with_constant_diagonal
sage: regular_symmetric_hadamard_matrix_with_constant_diagonal(4,1)
[ 1 1 1 -1]
[ 1 1 -1 1]
[ 1 -1 1 1]
[-1 1 1 1]
sage: regular_symmetric_hadamard_matrix_with_constant_diagonal(4,-1)
[ 1 -1 -1 -1]
[-1 1 -1 -1]
[-1 -1 1 -1]
[-1 -1 -1 1]
Other hardcoded values::
sage: for n,e in [(36,1),(36,-1),(100,1),(100,-1),(196, 1)]: # long time
....: print(repr(regular_symmetric_hadamard_matrix_with_constant_diagonal(n,e)))
36 x 36 dense matrix over Integer Ring
36 x 36 dense matrix over Integer Ring
100 x 100 dense matrix over Integer Ring
100 x 100 dense matrix over Integer Ring
196 x 196 dense matrix over Integer Ring
sage: for n,e in [(324,1),(324,-1)]: # not tested - long time, tested in RSHCD_324
....: print(repr(regular_symmetric_hadamard_matrix_with_constant_diagonal(n,e)))
324 x 324 dense matrix over Integer Ring
324 x 324 dense matrix over Integer Ring
From two close prime powers::
sage: regular_symmetric_hadamard_matrix_with_constant_diagonal(64,-1)
64 x 64 dense matrix over Integer Ring (use the '.str()' method to see the entries)
From a prime power and a conference matrix::
sage: regular_symmetric_hadamard_matrix_with_constant_diagonal(676,1) # long time
676 x 676 dense matrix over Integer Ring (use the '.str()' method to see the entries)
Recursive construction::
sage: regular_symmetric_hadamard_matrix_with_constant_diagonal(144,-1)
144 x 144 dense matrix over Integer Ring (use the '.str()' method to see the entries)
REFERENCE:
.. [BH12] \A. Brouwer and W. Haemers,
Spectra of graphs,
Springer, 2012,
http://homepages.cwi.nl/~aeb/math/ipm/ipm.pdf
.. [HX10] \W. Haemers and Q. Xiang,
Strongly regular graphs with parameters `(4m^4,2m^4+m^2,m^4+m^2,m^4+m^2)` exist for all `m>1`,
European Journal of Combinatorics,
Volume 31, Issue 6, August 2010, Pages 1553-1559,
:doi:`10.1016/j.ejc.2009.07.009`
"""
if existence and (n, e) in _rshcd_cache:
return _rshcd_cache[n, e]
from sage.graphs.strongly_regular_db import strongly_regular_graph
def true():
_rshcd_cache[n, e] = True
return True
M = None
if abs(e) != 1:
raise ValueError
sqn = None
if is_square(n):
sqn = int(sqrt(n))
if n < 0:
if existence:
return False
raise ValueError
elif n == 4:
if existence:
return true()
if e == 1:
M = J(4) - 2 * matrix(4, [[int(i + j == 3) for i in range(4)]
for j in range(4)])
else:
M = -J(4) + 2 * I(4)
elif n == 36:
if existence:
return true()
if e == 1:
M = strongly_regular_graph(36, 15, 6, 6).adjacency_matrix()
M = J(36) - 2 * M
else:
M = strongly_regular_graph(36, 14, 4, 6).adjacency_matrix()
M = -J(36) + 2 * M + 2 * I(36)
elif n == 100:
if existence:
return true()
if e == -1:
M = strongly_regular_graph(100, 44, 18, 20).adjacency_matrix()
M = 2 * M - J(100) + 2 * I(100)
else:
M = strongly_regular_graph(100, 45, 20, 20).adjacency_matrix()
M = J(100) - 2 * M
elif n == 196 and e == 1:
if existence:
return true()
M = strongly_regular_graph(196, 91, 42, 42).adjacency_matrix()
M = J(196) - 2 * M
elif n == 324:
if existence:
return true()
M = RSHCD_324(e)
elif (e == 1 and n % 16 == 0 and not sqn is None
and is_prime_power(sqn - 1) and is_prime_power(sqn + 1)):
if existence:
return true()
M = -rshcd_from_close_prime_powers(sqn)
elif (e == 1 and not sqn is None and sqn % 4 == 2
and True == strongly_regular_graph(sqn - 1, (sqn - 2) // 2,
(sqn - 6) // 4,
existence=True)
and is_prime_power(ZZ(sqn + 1))):
if existence:
return true()
M = rshcd_from_prime_power_and_conference_matrix(sqn + 1)
# Recursive construction: the kronecker product of two RSHCD is a RSHCD
else:
from itertools import product
for n1, e1 in product(divisors(n)[1:-1], [-1, 1]):
e2 = e1 * e
n2 = n // n1
if (regular_symmetric_hadamard_matrix_with_constant_diagonal(
n1, e1, existence=True) and
regular_symmetric_hadamard_matrix_with_constant_diagonal(
n2, e2, existence=True)):
if existence:
return true()
M1 = regular_symmetric_hadamard_matrix_with_constant_diagonal(
n1, e1)
M2 = regular_symmetric_hadamard_matrix_with_constant_diagonal(
n2, e2)
M = M1.tensor_product(M2)
break
if M is None:
from sage.misc.unknown import Unknown
_rshcd_cache[n, e] = Unknown
if existence:
return Unknown
raise ValueError("I do not know how to build a {}-RSHCD".format(
(n, e)))
assert M * M.transpose() == n * I(n)
assert set(map(sum, M)) == {ZZ(e * sqn)}
return M
