def cyclotomic_to_gamma(cyclo_up, cyclo_down):
"""
Convert a quotient of products of cyclotomic polynomials
to a quotient of products of polynomials `x^n - 1`.
INPUT:
- ``cyclo_up`` -- list of indices of cyclotomic polynomials in the numerator
- ``cyclo_down`` -- list of indices of cyclotomic polynomials in the denominator
OUTPUT:
a dictionary mapping an integer `n` to the power of `x^n - 1` that
appears in the given product
EXAMPLES::
sage: from sage.modular.hypergeometric_motive import cyclotomic_to_gamma
sage: cyclotomic_to_gamma([6], [1])
{2: -1, 3: -1, 6: 1}
"""
dico = defaultdict(int)
for d in cyclo_up:
dico[d] += 1
for d in cyclo_down:
dico[d] -= 1
resu = defaultdict(int)
for n in dico:
for d in divisors(n):
resu[d] += moebius(n / d) * dico[n]
return {d: resu[d] for d in resu if resu[d]}
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 range(n):
c.append(CuspFamily(N, d, label=str(i + 1)))
return c
def cyclotomic_to_gamma(cyclo_up, cyclo_down):
"""
Convert a quotient of products of cyclotomic polynomials
to a quotient of products of polynomials `x^n - 1`.
INPUT:
- ``cyclo_up`` -- list of indices of cyclotomic polynomials in the numerator
- ``cyclo_down`` -- list of indices of cyclotomic polynomials in the denominator
OUTPUT:
a dictionary mapping an integer `n` to the power of `x^n - 1` that
appears in the given product
EXAMPLES::
sage: from sage.modular.hypergeometric_motive import cyclotomic_to_gamma
sage: cyclotomic_to_gamma([6], [1])
{2: -1, 3: -1, 6: 1}
"""
dico = defaultdict(int)
for d in cyclo_up:
dico[d] += 1
for d in cyclo_down:
dico[d] -= 1
resu = defaultdict(int)
for n in dico:
for d in divisors(n):
resu[d] += moebius(n / d) * dico[n]
return {d: resu[d] for d in resu if resu[d]}
def origami_H2_2cyl_iterator(n, reducible=False, output="coordinates"):
r"""
INPUT:
- ``n`` - positive integer - the number of squares
- ``reducible`` - bool (default: False) - if True returns also the reducible
origamis.
- ``output`` - either "coordinates" or "origami" or "standard origami"
"""
for n1,n2 in OrderedPartitions(n,k=2):
for w1 in divisors(n1):
h1 = n1//w1
for w2 in filter(lambda x: x < w1, divisors(n2)):
h2 = n2//w2
if reducible or gcd(h1,h2) == 1:
d = gcd(w1,w2)
if d == 1:
for t1 in xrange(w1):
for t2 in xrange(w2):
if output == "coordinates":
yield w1,h1,t1,w2,h2,t2
elif output == "origami":
yield origami_H2_2cyl(w1,h1,t1,w2,h2,t2)
elif output == "standard_origami":
yield origami_H2_2cyl(w1,h1,t1,w2,h2,t2).standard_form()
else:
for t1 in xrange(w1):
for t2 in xrange(w2):
if reducible or gcd(d,h2*t1-h1*t2) == 1:
if output == "coordinates":
yield w1,h1,t1,w2,h2,t2
elif output == "origami":
yield origami_H2_2cyl(w1,h1,t1,w2,h2,t2)
elif output == "standard_origami":
yield origami_H2_2cyl(w1,h1,t1,w2,h2,t2).standard_form()
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]
"""
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)
dic = {
d: sum(bv[i] * long_etas[i].r(d) for i in range(r.nrows()))
for d in divisors(N)
}
short_etas.append(self(dic))
return short_etas
def _hermite_normal_forms(r, det):
"""
Generate all r x r integer matrices in (left) Hermite normal form
with the given determinant.
"""
if r == 0:
if det == 1:
yield matrix(ZZ, 0, [])
else:
for d in divisors(det):
for A in _hermite_normal_forms(r - 1, det // d):
for column in itertools.product(range(d), repeat=r - 1):
yield matrix(
ZZ, r, r, lambda i, j: A[i, j]
if i < r - 1 and j < r - 1 else d
if i == r - 1 and j == r - 1 else column[i]
if j == r - 1 else 0)
def capital_M(n):
"""
Auxiliary function, used to describe the canonical scheme.
INPUT:
- ``n`` -- an integer
OUTPUT:
a rational
EXAMPLES::
sage: from sage.modular.hypergeometric_motive import capital_M
sage: [capital_M(i) for i in range(1,8)]
[1, 4, 27, 64, 3125, 432, 823543]
"""
n = ZZ(n)
return QQ.prod(d ** (d * moebius(n / d)) for d in divisors(n))
def capital_M(n):
"""
Auxiliary function, used to describe the canonical scheme.
INPUT:
- ``n`` -- an integer
OUTPUT:
a rational
EXAMPLES::
sage: from sage.modular.hypergeometric_motive import capital_M
sage: [capital_M(i) for i in range(1,8)]
[1, 4, 27, 64, 3125, 432, 823543]
"""
n = ZZ(n)
return QQ.prod(d ** (d * moebius(n / d)) for d in divisors(n))
def DivisorLattice(n, facade=None):
"""
Return the divisor lattice of an integer.
Elements of the lattice are divisors of `n` and
`x < y` in the lattice if `x` divides `y`.
INPUT:
- ``n`` -- an integer
- ``facade`` (boolean) -- whether to make the returned poset a
facade poset (see :mod:`sage.categories.facade_sets`); the
default behaviour is the same as the default behaviour of
the :func:`~sage.combinat.posets.posets.Poset` constructor
EXAMPLES::
sage: P = Posets.DivisorLattice(12)
sage: sorted(P.cover_relations())
[[1, 2], [1, 3], [2, 4], [2, 6], [3, 6], [4, 12], [6, 12]]
sage: P = Posets.DivisorLattice(10, facade=False)
sage: P(2) < P(5)
False
TESTS::
sage: Posets.DivisorLattice(1)
Finite lattice containing 1 elements
"""
from sage.arith.misc import divisors
try:
n = Integer(n)
except TypeError:
raise TypeError(
"number of elements must be an integer, not {0}".format(n))
if n <= 0:
raise ValueError("n must be a positive integer")
return LatticePoset((divisors(n), lambda x, y: y % x == 0),
facade=facade,
linear_extension=True)
def DivisorLattice(n, facade=None):
"""
Return the divisor lattice of an integer.
Elements of the lattice are divisors of `n` and
`x < y` in the lattice if `x` divides `y`.
INPUT:
- ``n`` -- an integer
- ``facade`` (boolean) -- whether to make the returned poset a
facade poset (see :mod:`sage.categories.facade_sets`); the
default behaviour is the same as the default behaviour of
the :func:`~sage.combinat.posets.posets.Poset` constructor
EXAMPLES::
sage: P = Posets.DivisorLattice(12)
sage: sorted(P.cover_relations())
[[1, 2], [1, 3], [2, 4], [2, 6], [3, 6], [4, 12], [6, 12]]
sage: P = Posets.DivisorLattice(10, facade=False)
sage: P(2) < P(5)
False
TESTS::
sage: Posets.DivisorLattice(1)
Finite lattice containing 1 elements
"""
from sage.arith.misc import divisors
try:
n = Integer(n)
except TypeError:
raise TypeError("number of elements must be an integer, not {0}".format(n))
if n <= 0:
raise ValueError("n must be a positive integer")
return LatticePoset( (divisors(n), lambda x, y: y % x == 0),
facade=facade, linear_extension=True)
def gamma_list_to_cyclotomic(galist):
r"""
Convert a quotient of products of polynomials `x^n - 1`
to a quotient of products of cyclotomic polynomials.
INPUT:
- ``galist`` -- a list of integers, where an integer `n` represents
the power `(x^{|n|} - 1)^{\operatorname{sgn}(n)}`
OUTPUT:
a pair of list of integers, where `k` represents the cyclotomic
polynomial `\Phi_k`
EXAMPLES::
sage: from sage.modular.hypergeometric_motive import gamma_list_to_cyclotomic
sage: gamma_list_to_cyclotomic([-1, -1, 2])
([2], [1])
sage: gamma_list_to_cyclotomic([-1, -1, -1, -3, 6])
([2, 6], [1, 1, 1])
sage: gamma_list_to_cyclotomic([-1, 2, 3, -4])
([3], [4])
sage: gamma_list_to_cyclotomic([8,2,2,2,-6,-4,-3,-1])
([2, 2, 8], [3, 3, 6])
"""
resu = defaultdict(int)
for n in galist:
eps = sgn(n)
for d in divisors(abs(n)):
resu[d] += eps
return (sorted(d for d in resu for k in range(resu[d])),
sorted(d for d in resu for k in range(-resu[d])))
def origami_H2_1cyl_iterator(n, reducible=False, output='coordinates'):
r"""
INPUT:
- ``n`` - positive integer - the number of squares
- ``reducible`` - bool (default: False) - if True returns also the reducible
origamis.
- ``output`` - either "coordinates" or "origami" or "standard origami"
"""
if reducible:
hlist = divisors(n)
else:
hlist = [1]
for h in hlist:
for p in Partitions(Integer(n/h),length=3):
for l in CyclicPermutationsOfPartition([p]):
l1,l2,l3 = l[0]
if l1 == l2 and l2 == l3:
if reducible or l1 == 1:
for t in xrange(l1):
if output == "coordinates":
yield l1,l2,l3,h,t
elif output == "origami":
yield origami_H2_1cyl(l1,l2,l3,h,t)
elif output == "standard_origami":
yield origami_H2_1cyl(l1,l2,l3,h,t).standard_form()
elif reducible or gcd([l1,l2,l3]) == 1:
for t in xrange(n):
if output == "coordinates":
yield l1,l2,l3,h,t
elif output == "origami":
yield origami_H2_1cyl(l1,l2,l3,h,t)
elif output == "standard_origami":
yield origami_H2_1cyl(l1,l2,l3,h,t).standard_form()
def gamma_list_to_cyclotomic(galist):
r"""
Convert a quotient of products of polynomials `x^n - 1`
to a quotient of products of cyclotomic polynomials.
INPUT:
- ``galist`` -- a list of integers, where an integer `n` represents
the power `(x^{|n|} - 1)^{\operatorname{sgn}(n)}`
OUTPUT:
a pair of list of integers, where `k` represents the cyclotomic
polynomial `\Phi_k`
EXAMPLES::
sage: from sage.modular.hypergeometric_motive import gamma_list_to_cyclotomic
sage: gamma_list_to_cyclotomic([-1, -1, 2])
([2], [1])
sage: gamma_list_to_cyclotomic([-1, -1, -1, -3, 6])
([2, 6], [1, 1, 1])
sage: gamma_list_to_cyclotomic([-1, 2, 3, -4])
([3], [4])
sage: gamma_list_to_cyclotomic([8,2,2,2,-6,-4,-3,-1])
([2, 2, 8], [3, 3, 6])
"""
resu = defaultdict(int)
for n in galist:
eps = sgn(n)
for d in divisors(abs(n)):
resu[d] += eps
return (sorted(d for d in resu for k in range(resu[d])),
sorted(d for d in resu for k in range(-resu[d])))
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)^-3 (eta_2)^2 (eta_3)^1 (eta_4)^-1 (eta_6)^-2 (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)^6 (eta_2)^-9 (eta_3)^-2 (eta_4)^3 (eta_6)^3 (eta_12)^-1,
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)^3 (eta_3)^-1 (eta_4)^-3 (eta_12)^1]
sage: EtaGroup(12).basis(reduce=False) # much bigger coefficients
[Eta product of level 12 : (eta_1)^384 (eta_2)^-576 (eta_3)^-696 (eta_4)^216 (eta_6)^576 (eta_12)^96,
Eta product of level 12 : (eta_2)^24 (eta_12)^-24,
Eta product of level 12 : (eta_1)^-40 (eta_2)^116 (eta_3)^96 (eta_4)^-30 (eta_6)^-80 (eta_12)^-62,
Eta product of level 12 : (eta_1)^-4 (eta_2)^-33 (eta_3)^-4 (eta_4)^1 (eta_6)^3 (eta_12)^37,
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 di in divs:
# generate a row of relation matrix
row = [Mod(di, 24) - Mod(N, 24), Mod(N // di, 24) - Mod(1, 24)]
for p in primedivs:
row.append(Mod(12 * (N // di).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 weight_two_basis_from_theta_blocks(N,
prec,
dim,
jacobiforms=None,
verbose=False):
r"""
Look for theta blocks of weight two and given index among the infinite families of weight two theta blocks associated to the root systems A_4, B_2+G_2, A_1+B_3, A_1+C_3
This is not meant to be called directly. Use JacobiForms(N).basis(2) with the optional parameter try_theta_blocks = True.
INPUT:
- ``N`` -- the index
- ``prec`` -- precision
- ``dim`` -- the dimension of the space of Jacobi forms.
- ``jacobiforms`` -- a JacobiForms instance for this index. (If none is provided then we construct one now.)
- ``verbose`` -- boolean (default False); if True then we comment on the computations.
OUTPUT: a list of JacobiForm's
"""
if not jacobiforms:
jacobiforms = JacobiForms(N)
rank = 0
#the four families of weight two theta blocks from root lattices
thetablockQ_1 = QuadraticForm(
matrix([[4, 3, 2, 1], [3, 6, 4, 2], [2, 4, 6, 3], [1, 2, 3, 4]]))
thetablockQ_2 = QuadraticForm(
matrix([[24, 12, 0, 0], [12, 8, 0, 0], [0, 0, 12, 6], [0, 0, 6, 6]]))
thetablockQ_3 = QuadraticForm(
matrix([[4, 0, 0, 0], [0, 20, 10, 20], [0, 10, 10, 20],
[0, 20, 20, 60]]))
thetablockQ_4 = QuadraticForm(
matrix([[4, 0, 0, 0], [0, 8, 8, 4], [0, 8, 16, 8], [0, 4, 8, 6]]))
thetablock_tuple = thetablockQ_1, thetablockQ_2, thetablockQ_3, thetablockQ_4
from .jacobi_forms_class import theta_block
thetablock_1 = lambda a, b, c, d, prec: theta_block(
[a, a + b, a + b + c, a + b + c + d, b, b + c, b + c + d, c, c + d, d],
-6,
prec,
jacobiforms=jacobiforms)
thetablock_2 = lambda a, b, c, d, prec: theta_block([
a, 3 * a + b, 3 * a + b + b, a + a + b, a + b, b, c + c, c + c + d, 2 *
(c + d), d
],
-6,
prec,
jacobiforms=jacobiforms
)
thetablock_3 = lambda a, b, c, d, prec: theta_block([
a + a, b + b, b + b + c, 2 * (b + c + d + d), b + b + c + d + d, b + b
+ c + 4 * d, c, c + d + d, c + 4 * d, d + d
],
-6,
prec,
jacobiforms=jacobiforms
)
thetablock_4 = lambda a, b, c, d, prec: theta_block([
a + a, b, b + b + c + c + d, b + c, b + c + c + d, b + c + d, c, c + c
+ d, c + d, d
],
-6,
prec,
jacobiforms=jacobiforms
)
thetablocks = thetablock_1, thetablock_2, thetablock_3, thetablock_4
basis = []
basis_vectors = []
pivots = []
div_N = divisors(N)
div_N.reverse()
for i, Q in enumerate(thetablock_tuple):
v_list = Q.short_vector_list_up_to_length(N + 1, up_to_sign_flag=True)
if verbose:
print(
'I am looking through the theta block family of the root system %s.'
% ['A_4', 'B_2+G_2', 'A_1+B_3', 'A_1+C_3'][i])
for _d in div_N:
prec_d = prec * (N // _d)
for v in v_list[_d]:
a, b, c, d = v
try:
j = thetablocks[i](a, b, c, d, prec_d).hecke_V(N // _d)
old_rank = 0 + rank
basis, basis_vectors, pivots, rank = update_echelon_form_with(
j, basis, basis_vectors, pivots, rank, sage_one_eighth)
if verbose and old_rank < rank:
if i == 0:
L = [
abs(x)
for x in (a, a + b, a + b + c, a + b + c + d,
b, b + c, b + c + d, c, c + d, d)
]
elif i == 1:
L = [
abs(x) for x in (a, 3 * a + b, 3 * a + b + b,
a + a + b, a + b, b, c + c,
c + c + d, 2 * (c + d), d)
]
elif i == 2:
L = [
abs(x)
for x in (a + a, b + b, b + b + c,
2 * (b + c + d + d),
b + b + c + d + d, b + b + c + 4 * d,
c, c + d + d, c + 4 * d, d + d)
]
else:
L = [
abs(x)
for x in (a + a, b, b + b + c + c + d, b + c,
b + c + c + d, b + c + d, c,
c + c + d, c + d, d)
]
L.sort()
print('I found the theta block Th_' + str(L) +
' / eta^6.')
if rank == dim:
return basis
except ValueError:
pass
if verbose:
print(
'I did not find enough theta blocks. Time to try something else.')
return jacobiforms.basis(2, prec, try_theta_blocks=False, verbose=verbose)
def weight_three_basis_from_theta_blocks(N,
prec,
dim,
jacobiforms=None,
verbose=False):
r"""
Look for theta blocks of weight three and given index among the infinite families of weight three theta blocks associated to the root systems B_3, C_3, A_2+A_3, 3A_2, 2A_1 + A_2 + B_2, 3A_1 + A_3, A_6, A_1+D_5.
This is not meant to be called directly. Use JacobiForms(N).basis(3) with the optional parameter try_theta_blocks = True.
INPUT:
- ``N`` -- the index
- ``prec`` -- precision
- ``dim`` -- the dimension of the space of Jacobi forms.
- ``jacobiforms`` -- a JacobiForms instance for this index. (If none is provided then we construct one now.)
- ``verbose`` -- boolean (default False); if True then we comment on the computations.
OUTPUT: a list of JacobiForm's
"""
if not jacobiforms:
jacobiforms = JacobiForms(N)
rank = 0
thetablockQ_1 = QuadraticForm(
matrix([[20, 10, 20], [10, 10, 20], [20, 20, 60]])) #B3
thetablockQ_2 = QuadraticForm(matrix([[8, 8, 4], [8, 16, 8], [4, 8,
6]])) #C3
thetablockQ_3 = QuadraticForm(
matrix([[2, 1, 0, 0, 0], [1, 2, 0, 0, 0], [0, 0, 12, 4, 4],
[0, 0, 4, 4, 4], [0, 0, 4, 4, 12]])) #A2 + A3
thetablockQ_4 = QuadraticForm(
matrix([[2, 1, 0, 0, 0, 0], [1, 2, 0, 0, 0, 0], [0, 0, 2, 1, 0, 0],
[0, 0, 1, 2, 0, 0], [0, 0, 0, 0, 2, 1], [0, 0, 0, 0, 1,
2]])) #3 A2
thetablockQ_5 = QuadraticForm(
matrix([[4, 0, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0], [0, 0, 2, 1, 0, 0],
[0, 0, 1, 2, 0, 0], [0, 0, 0, 0, 6, 6], [0, 0, 0, 0, 6,
12]])) #2A1 + A2 + B2
thetablockQ_6 = QuadraticForm(
matrix([[4, 0, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0], [0, 0, 4, 0, 0, 0],
[0, 0, 0, 12, 4, 4], [0, 0, 0, 4, 4, 4], [0, 0, 0, 4, 4,
12]])) #3A1 + A3
thetablockQ_7 = QuadraticForm(
matrix([[6, 5, 4, 3, 2, 1], [5, 10, 8, 6, 4, 2], [4, 8, 12, 9, 6, 3],
[3, 6, 9, 12, 8, 4], [2, 4, 6, 8, 10, 5], [1, 2, 3, 4, 5,
6]])) #A6
thetablockQ_8 = QuadraticForm(
matrix([[4, 0, 0, 0, 0, 0], [0, 10, 6, 12, 8, 4], [0, 6, 10, 12, 8, 4],
[0, 12, 12, 24, 16, 8], [0, 8, 8, 16, 16, 8],
[0, 4, 4, 8, 8, 8]])) #A1 + D5
thetablock_tuple = thetablockQ_1, thetablockQ_2, thetablockQ_3, thetablockQ_4, thetablockQ_5, thetablockQ_6, thetablockQ_7, thetablockQ_8
args1 = lambda b, c, d: [
b + b, b + b + c, 2 * (b + c + d + d), b + b + c + d + d, b + b + c + 4
* d, c, c + d + d, c + 4 * d, d + d
]
args2 = lambda b, c, d: [
b, b + b + c + c + d, b + c, b + c + c + d, b + c + d, c, c + c + d, c
+ d, d
]
args3 = lambda a, b, d, e, f: [
a, a + b, b, d + d, e, d + d + e, f + f, e + f + f, d + d + e + f + f
]
args4 = lambda a, b, c, d, e, f: [a, a + b, b, c, c + d, d, e, e + f, f]
args5 = lambda a, b, c, d, e, f: [
a + a, b + b, c, c + d, d, e, f + f, e + f + f, e + e + f + f
]
args6 = lambda a, b, c, d, e, f: [
a + a, b + b, c + c, d + d, e, d + d + e, f + f, e + f + f, d + d + e +
f + f
]
args7 = lambda a, b, c, d, e, f: [
a, a + b, a + b + c, a + b + c + d, a + b + c + d + e, a + b + c + d +
e + f, b, b + c, b + c + d, b + c + d + e, b + c + d + e + f, c, c + d,
c + d + e, c + d + e + f, d, d + e, d + e + f, e, e + f, f
]
args8 = lambda a, b, c, d, e, f: [
a + a, b, c, d, b + d, c + d, b + c + d, e, d + e, b + d + e, c + d +
e, b + c + d + e, b + c + 2 * d + e, f, e + f, d + e + f, b + d + e +
f, c + d + e + f, b + c + d + e + f, b + c + 2 * d + e + f, b + c + 2 *
d + 2 * e + f
]
args_tuple = args1, args2, args3, args4, args5, args6, args7, args8
from .jacobi_forms_class import theta_block
thetablock_1 = lambda b, c, d, prec: theta_block(
args1(b, c, d), -3, prec, jacobiforms=jacobiforms)
thetablock_2 = lambda b, c, d, prec: theta_block(
args2(b, c, d), -3, prec, jacobiforms=jacobiforms)
thetablock_3 = lambda a, b, d, e, f, prec: theta_block(
args3(a, b, d, e, f), -3, prec, jacobiforms=jacobiforms)
thetablock_4 = lambda a, b, c, d, e, f, prec: theta_block(
args4(a, b, c, d, e, f), -3, prec, jacobiforms=jacobiforms)
thetablock_5 = lambda a, b, c, d, e, f, prec: theta_block(
args5(a, b, c, d, e, f), -3, prec, jacobiforms=jacobiforms)
thetablock_6 = lambda a, b, c, d, e, f, prec: theta_block(
args6(a, b, c, d, e, f), -3, prec, jacobiforms=jacobiforms)
thetablock_7 = lambda a, b, c, d, e, f, prec: theta_block(
args7(a, b, c, d, e, f), -15, prec, jacobiforms=jacobiforms)
thetablock_8 = lambda a, b, c, d, e, f, prec: theta_block(
args8(a, b, c, d, e, f), -15, prec, jacobiforms=jacobiforms)
thetablocks = thetablock_1, thetablock_2, thetablock_3, thetablock_4, thetablock_5, thetablock_6, thetablock_7, thetablock_8
basis = []
basis_vectors = []
pivots = []
div_N = divisors(N)
div_N.reverse()
for i, Q in enumerate(thetablock_tuple):
v_list = Q.short_vector_list_up_to_length(N + 1, up_to_sign_flag=True)
if verbose:
print(
'I am looking through the theta block family of the root system %s.'
% [
'B_3', 'C_3', 'A_2+A_3', '3A_2', '2A_1+A_2+B_2',
'3A_1 + A_3', 'A_6', 'A_1+D_5'
][i])
for _d in div_N:
prec_d = prec * (N // _d)
for v in v_list[_d]:
if all(a for a in args_tuple[i](*v)):
try:
j = thetablocks[i](*(list(v) + [prec])).hecke_V(N //
_d)
old_rank = 0 + rank
basis, basis_vectors, pivots, rank = update_echelon_form_with(
j, basis, basis_vectors, pivots, rank,
sage_one_eighth)
if verbose and old_rank < rank:
L = [abs(x) for x in args_tuple[i](*v)]
L.sort()
if _d == N:
print('I found the theta block Th_' + str(L) +
[' / eta^3.', ' / eta^15.'][i >= 6])
else:
print(
'I applied the Hecke operator V_%d to the theta block Th_'
% (N // _d) + str(L) +
[' / eta^3.', ' / eta^15.'][i >= 6])
if rank == dim:
return basis
except ValueError:
pass
if verbose:
print(
'I did not find enough theta blocks. Time to try something else.')
return jacobiforms.basis(2, prec, try_theta_blocks=False, verbose=verbose)