def _coefficient_by_restriction__with_restriction_matrix( precision, k, relation_precision, global_restriction_matrix__big, row_groups, row_labels, column_labels, global_relation_matrix, column_labels_relations, dim = None ) :
r"""
Compute the Fourier expansions of Jacobi forms (over `\QQ`) of weight `k` and
index `L` (an even symmetric matrix) up to given precision.
NOTE:
In case ``relation_precision`` is not ``None``, we assume that it is of the form
`\{ (n, r) : n < B \}`, where `B` is the index of the filter.
INPUT:
- ``precision`` -- A filter for Jacobi forms of arbitrary index.
- `k` -- An integer.
- ``relation_precision`` -- A filter for Jacobi forms or ``None`` (default: ``None``).
OUTPUT:
- A list of elements of the corresponding Fourier expansion module.
TESTS:
See ``_test__coefficient_by_restriction`` for further tests.
::
sage: from psage.modform.jacobiforms.jacobiformd1_fourierexpansion import *
sage: from psage.modform.jacobiforms.jacobiformd1_fegenerators import _coefficient_by_restriction
sage: indices = JacobiFormD1Indices(QuadraticForm(matrix(2, [2,1,1,2])))
sage: precision = indices.filter(20)
sage: relation_precision = indices.filter(10)
sage: _coefficient_by_restriction(precision, 10) == _coefficient_by_restriction(precision, 10, relation_precision)
True
"""
L = precision.jacobi_index()
Lmat = L.matrix()
Lmat.set_immutable()
if dim is None :
dim = dimension__jacobi(k, Lmat)
if relation_precision == precision :
assert column_labels == column_labels_relations
if column_labels != column_labels_relations :
row_groups__small = [ len(filter( lambda (n,r): n < relation_precision.index(), row_labels[m].keys())) for (_, m, _, _) in row_groups ]
def _coefficient_by_restriction( precision, k, relation_precision = None ) :
r"""
Compute the Fourier expansions of Jacobi forms (over `\QQ`) of weight `k` and
index `L` (an even symmetric matrix) up to given precision.
ALGORITHM:
See [Raum, Computing vector valued modular forms and Jacobi forms]. The algorithm will be applied for precision ``relation_precision``.
The remaining Fourier coefficients will be determined using fewer restrictions.
NOTE:
In case ``relation_precision`` is not ``None``, we assume that it is of the form
`\{ (n, r) : n < B \}`, where `B` is the index of the filter.
INPUT:
- ``precision`` -- A filter for Jacobi forms of arbitrary index.
- `k` -- An integer.
- ``relation_precision`` -- A filter for Jacobi forms or ``None`` (default: ``None``).
OUTPUT:
- A list of elements of the corresponding Fourier expansion module.
TESTS:
See ``_test__coefficient_by_restriction`` for further tests.
::
sage: from psage.modform.jacobiforms.jacobiformd1_fourierexpansion import *
sage: from psage.modform.jacobiforms.jacobiformd1_fegenerators import _coefficient_by_restriction
sage: indices = JacobiFormD1Indices(QuadraticForm(matrix(2, [2,1,1,2])))
sage: precision = indices.filter(20)
sage: relation_precision = indices.filter(10)
sage: _coefficient_by_restriction(precision, 10) == _coefficient_by_restriction(precision, 10, relation_precision)
True
"""
L = precision.jacobi_index()
Lmat = L.matrix()
Lmat.set_immutable()
global _coefficient_by_restriction__cache
try :
expansions = _coefficient_by_restriction__cache[(k, Lmat)]
if len(expansions) == 0 \
or expansions[0].precision() >= precision :
## TODO: This should be checked in the framework
return [ f.truncate(precision) for f in expansions ]
except KeyError :
pass
dim = dimension__jacobi(k, Lmat)
if dim == 0 :
return []
if relation_precision is None :
relation_precision = precision
R = precision.monoid()._r_representatives
S_extended = _find_complete_set_of_restriction_vectors(L, R, reduction_function = precision.monoid().reduce_r)
S = list()
for (s, _) in S_extended :
if s not in S :
S.append(s)
max_S_length = max([L(s) for s in S])
B_correction = 1 + (k + max_S_length) // 12
if precision.index() < B_correction :
precision = precision.monoid().filter(B_correction)
if relation_precision.index() < B_correction :
relation_precision = precision.monoid().filter(B_correction)
(global_restriction_matrix__big, row_groups, row_labels, column_labels, global_relation_matrix, column_labels_relations) = \
_prepare_coefficient_by_restriction(precision, k, relation_precision, S)
expansions = _coefficient_by_restriction__with_restriction_matrix( precision, k, relation_precision, global_restriction_matrix__big,
row_groups, row_labels, column_labels, global_relation_matrix, column_labels_relations, dim )
_coefficient_by_restriction__cache[(k, Lmat)] = expansions
return expansions
def _coefficient_by_restriction__with_restriction_matrix(
precision,
k,
relation_precision,
global_restriction_matrix__big,
row_groups,
row_labels,
column_labels,
global_relation_matrix,
column_labels_relations,
dim=None):
r"""
Compute the Fourier expansions of Jacobi forms (over `\QQ`) of weight `k` and
index `L` (an even symmetric matrix) up to given precision.
NOTE:
In case ``relation_precision`` is not ``None``, we assume that it is of the form
`\{ (n, r) : n < B \}`, where `B` is the index of the filter.
INPUT:
- ``precision`` -- A filter for Jacobi forms of arbitrary index.
- `k` -- An integer.
- ``relation_precision`` -- A filter for Jacobi forms or ``None`` (default: ``None``).
OUTPUT:
- A list of elements of the corresponding Fourier expansion module.
TESTS:
See ``_test__coefficient_by_restriction`` for further tests.
::
sage: from psage.modform.jacobiforms.jacobiformd1_fourierexpansion import *
sage: from psage.modform.jacobiforms.jacobiformd1_fegenerators import _coefficient_by_restriction
sage: indices = JacobiFormD1Indices(QuadraticForm(matrix(2, [2,1,1,2])))
sage: precision = indices.filter(20)
sage: relation_precision = indices.filter(10)
sage: _coefficient_by_restriction(precision, 10) == _coefficient_by_restriction(precision, 10, relation_precision)
True
"""
L = precision.jacobi_index()
Lmat = L.matrix()
Lmat.set_immutable()
if dim is None:
dim = dimension__jacobi(k, Lmat)
if relation_precision == precision:
assert column_labels == column_labels_relations
if column_labels != column_labels_relations:
row_groups__small = [
len(
filter(lambda (n, r): n < relation_precision.index(),
row_labels[m].keys())) for (_, m, _, _) in row_groups
]
def _coefficient_by_restriction(precision, k, relation_precision=None):
r"""
Compute the Fourier expansions of Jacobi forms (over `\QQ`) of weight `k` and
index `L` (an even symmetric matrix) up to given precision.
ALGORITHM:
See [Raum, Computing vector valued modular forms and Jacobi forms]. The algorithm will be applied for precision ``relation_precision``.
The remaining Fourier coefficients will be determined using fewer restrictions.
NOTE:
In case ``relation_precision`` is not ``None``, we assume that it is of the form
`\{ (n, r) : n < B \}`, where `B` is the index of the filter.
INPUT:
- ``precision`` -- A filter for Jacobi forms of arbitrary index.
- `k` -- An integer.
- ``relation_precision`` -- A filter for Jacobi forms or ``None`` (default: ``None``).
OUTPUT:
- A list of elements of the corresponding Fourier expansion module.
TESTS:
See ``_test__coefficient_by_restriction`` for further tests.
::
sage: from psage.modform.jacobiforms.jacobiformd1_fourierexpansion import *
sage: from psage.modform.jacobiforms.jacobiformd1_fegenerators import _coefficient_by_restriction
sage: indices = JacobiFormD1Indices(QuadraticForm(matrix(2, [2,1,1,2])))
sage: precision = indices.filter(20)
sage: relation_precision = indices.filter(10)
sage: _coefficient_by_restriction(precision, 10) == _coefficient_by_restriction(precision, 10, relation_precision)
True
"""
L = precision.jacobi_index()
Lmat = L.matrix()
Lmat.set_immutable()
global _coefficient_by_restriction__cache
try:
expansions = _coefficient_by_restriction__cache[(k, Lmat)]
if len(expansions) == 0 \
or expansions[0].precision() >= precision :
## TODO: This should be checked in the framework
return [f.truncate(precision) for f in expansions]
except KeyError:
pass
dim = dimension__jacobi(k, Lmat)
if dim == 0:
return []
if relation_precision is None:
relation_precision = precision
R = precision.monoid()._r_representatives
S_extended = _find_complete_set_of_restriction_vectors(
L, R, reduction_function=precision.monoid().reduce_r)
S = list()
for (s, _) in S_extended:
if s not in S:
S.append(s)
max_S_length = max([L(s) for s in S])
B_correction = 1 + (k + max_S_length) // 12
if precision.index() < B_correction:
precision = precision.monoid().filter(B_correction)
if relation_precision.index() < B_correction:
relation_precision = precision.monoid().filter(B_correction)
(global_restriction_matrix__big, row_groups, row_labels, column_labels, global_relation_matrix, column_labels_relations) = \
_prepare_coefficient_by_restriction(precision, k, relation_precision, S)
expansions = _coefficient_by_restriction__with_restriction_matrix(
precision, k, relation_precision, global_restriction_matrix__big,
row_groups, row_labels, column_labels, global_relation_matrix,
column_labels_relations, dim)
_coefficient_by_restriction__cache[(k, Lmat)] = expansions
return expansions
def _rank(self, K) :
if K is QQ or K in NumberFields() :
return dimension__jacobi( self.__weight, QuadraticForm(matrix(1, [2 * self.__index])) )
raise NotImplementedError
def _rank(self, K) :
return dimension__jacobi(self.__weight, self.__index)
