def _delta_poly(prec=10):
"""
Return the q-expansion of Delta as a FLINT polynomial. Used internally by
the :func:`~delta_qexp` function. See the docstring of :func:`~delta_qexp`
for more information.
INPUT:
- ``prec`` -- integer; the absolute precision of the output
OUTPUT:
the q-expansion of Delta to precision ``prec``, as a FLINT
:class:`~sage.libs.flint.fmpz_poly.Fmpz_poly` object.
EXAMPLES::
sage: from sage.modular.modform.vm_basis import _delta_poly
sage: _delta_poly(7)
7 0 1 -24 252 -1472 4830 -6048
"""
if prec <= 0:
raise ValueError("prec must be positive")
v = [0] * prec
# Let F = \sum_{n >= 0} (-1)^n (2n+1) q^(floor(n(n+1)/2)).
# Then delta is F^8.
# First compute F^2 directly by naive polynomial multiplication,
# since F is very sparse.
stop = int((-1 + math.sqrt(1 + 8 * prec)) / 2.0)
# make list of index/value pairs for the sparse poly
values = [(n*(n+1)//2, ((-2*n-1) if (n & 1) else (2*n+1))) \
for n in xrange(stop+1)]
for (i1, v1) in values:
for (i2, v2) in values:
try:
v[i1 + i2] += v1 * v2
except IndexError:
break
f = Fmpz_poly(v)
t = verbose('made series')
f = f * f
f._unsafe_mutate_truncate(prec)
t = verbose('squared (2 of 3)', t)
f = f * f
f._unsafe_mutate_truncate(prec - 1)
t = verbose('squared (3 of 3)', t)
f = f.left_shift(1)
t = verbose('shifted', t)
return f
def _delta_poly(prec=10):
"""
Return the q-expansion of Delta as a FLINT polynomial. Used internally by
the :func:`~delta_qexp` function. See the docstring of :func:`~delta_qexp`
for more information.
INPUT:
- ``prec`` -- integer; the absolute precision of the output
OUTPUT:
the q-expansion of Delta to precision ``prec``, as a FLINT
:class:`~sage.libs.flint.fmpz_poly.Fmpz_poly` object.
EXAMPLES::
sage: from sage.modular.modform.vm_basis import _delta_poly
sage: _delta_poly(7)
7 0 1 -24 252 -1472 4830 -6048
"""
if prec <= 0:
raise ValueError("prec must be positive")
v = [0] * prec
# Let F = \sum_{n >= 0} (-1)^n (2n+1) q^(floor(n(n+1)/2)).
# Then delta is F^8.
# First compute F^2 directly by naive polynomial multiplication,
# since F is very sparse.
stop = int((-1+math.sqrt(1+8*prec))/2.0)
# make list of index/value pairs for the sparse poly
values = [(n*(n+1)//2, ((-2*n-1) if (n & 1) else (2*n+1))) \
for n in xrange(stop+1)]
for (i1, v1) in values:
for (i2, v2) in values:
try:
v[i1 + i2] += v1 * v2
except IndexError:
break
f = Fmpz_poly(v)
t = verbose('made series')
f = f*f
f._unsafe_mutate_truncate(prec)
t = verbose('squared (2 of 3)', t)
f = f*f
f._unsafe_mutate_truncate(prec - 1)
t = verbose('squared (3 of 3)', t)
f = f.left_shift(1)
t = verbose('shifted', t)
return f
def victor_miller_basis(k, prec=10, cusp_only=False, var='q'):
r"""
Compute and return the Victor Miller basis for modular forms of
weight `k` and level 1 to precision `O(q^{prec})`. If
``cusp_only`` is True, return only a basis for the cuspidal
subspace.
INPUT:
- ``k`` -- an integer
- ``prec`` -- (default: 10) a positive integer
- ``cusp_only`` -- bool (default: False)
- ``var`` -- string (default: 'q')
OUTPUT:
A sequence whose entries are power series in ``ZZ[[var]]``.
EXAMPLES::
sage: victor_miller_basis(1, 6)
[]
sage: victor_miller_basis(0, 6)
[
1 + O(q^6)
]
sage: victor_miller_basis(2, 6)
[]
sage: victor_miller_basis(4, 6)
[
1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + O(q^6)
]
sage: victor_miller_basis(6, 6, var='w')
[
1 - 504*w - 16632*w^2 - 122976*w^3 - 532728*w^4 - 1575504*w^5 + O(w^6)
]
sage: victor_miller_basis(6, 6)
[
1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 + O(q^6)
]
sage: victor_miller_basis(12, 6)
[
1 + 196560*q^2 + 16773120*q^3 + 398034000*q^4 + 4629381120*q^5 + O(q^6),
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)
]
sage: victor_miller_basis(12, 6, cusp_only=True)
[
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)
]
sage: victor_miller_basis(24, 6, cusp_only=True)
[
q + 195660*q^3 + 12080128*q^4 + 44656110*q^5 + O(q^6),
q^2 - 48*q^3 + 1080*q^4 - 15040*q^5 + O(q^6)
]
sage: victor_miller_basis(24, 6)
[
1 + 52416000*q^3 + 39007332000*q^4 + 6609020221440*q^5 + O(q^6),
q + 195660*q^3 + 12080128*q^4 + 44656110*q^5 + O(q^6),
q^2 - 48*q^3 + 1080*q^4 - 15040*q^5 + O(q^6)
]
sage: victor_miller_basis(32, 6)
[
1 + 2611200*q^3 + 19524758400*q^4 + 19715347537920*q^5 + O(q^6),
q + 50220*q^3 + 87866368*q^4 + 18647219790*q^5 + O(q^6),
q^2 + 432*q^3 + 39960*q^4 - 1418560*q^5 + O(q^6)
]
sage: victor_miller_basis(40,200)[1:] == victor_miller_basis(40,200,cusp_only=True)
True
sage: victor_miller_basis(200,40)[1:] == victor_miller_basis(200,40,cusp_only=True)
True
AUTHORS:
- William Stein, Craig Citro: original code
- Martin Raum (2009-08-02): use FLINT for polynomial arithmetic (instead of NTL)
"""
k = Integer(k)
if k%2 == 1 or k==2:
return Sequence([])
elif k < 0:
raise ValueError("k must be non-negative")
elif k == 0:
return Sequence([PowerSeriesRing(ZZ,var)(1).add_bigoh(prec)], cr=True)
e = k.mod(12)
if e == 2: e += 12
n = (k-e) // 12
if n == 0 and cusp_only:
return Sequence([])
# If prec is less than or equal to the dimension of the space of
# cusp forms, which is just n, then we know the answer, and we
# simply return it.
if prec <= n:
q = PowerSeriesRing(ZZ,var).gen(0)
err = bigO(q**prec)
ls = [0] * (n+1)
if not cusp_only:
ls[0] = 1 + err
for i in range(1,prec):
ls[i] = q**i + err
for i in range(prec,n+1):
ls[i] = err
return Sequence(ls, cr=True)
F6 = eisenstein_series_poly(6,prec)
if e == 0:
A = Fmpz_poly(1)
elif e == 4:
A = eisenstein_series_poly(4,prec)
elif e == 6:
A = F6
elif e == 8:
A = eisenstein_series_poly(8,prec)
elif e == 10:
A = eisenstein_series_poly(10,prec)
else: # e == 14
A = eisenstein_series_poly(14,prec)
if A[0] == -1 :
A = -A
if n == 0:
return Sequence([PowerSeriesRing(ZZ,var)(A.list()).add_bigoh(prec)],cr=True)
F6_squared = F6**2
F6_squared._unsafe_mutate_truncate(prec)
D = _delta_poly(prec)
Fprod = F6_squared
Dprod = D
if cusp_only:
ls = [Fmpz_poly(0)] + [A] * n
else:
ls = [A] * (n+1)
for i in xrange(1,n+1):
ls[n-i] *= Fprod
ls[i] *= Dprod
ls[n-i]._unsafe_mutate_truncate(prec)
ls[i]._unsafe_mutate_truncate(prec)
Fprod *= F6_squared
Dprod *= D
Fprod._unsafe_mutate_truncate(prec)
Dprod._unsafe_mutate_truncate(prec)
P = PowerSeriesRing(ZZ,var)
if cusp_only :
for i in xrange(1,n+1) :
for j in xrange(1, i) :
ls[j] = ls[j] - ls[j][i]*ls[i]
return Sequence(map(lambda l: P(l.list()).add_bigoh(prec), ls[1:]),cr=True)
else :
for i in xrange(1,n+1) :
for j in xrange(i) :
ls[j] = ls[j] - ls[j][i]*ls[i]
return Sequence(map(lambda l: P(l.list()).add_bigoh(prec), ls), cr=True)
def victor_miller_basis(k, prec=10, cusp_only=False, var='q'):
r"""
Compute and return the Victor Miller basis for modular forms of
weight `k` and level 1 to precision `O(q^{prec})`. If
``cusp_only`` is True, return only a basis for the cuspidal
subspace.
INPUT:
- ``k`` -- an integer
- ``prec`` -- (default: 10) a positive integer
- ``cusp_only`` -- bool (default: False)
- ``var`` -- string (default: 'q')
OUTPUT:
A sequence whose entries are power series in ``ZZ[[var]]``.
EXAMPLES::
sage: victor_miller_basis(1, 6)
[]
sage: victor_miller_basis(0, 6)
[
1 + O(q^6)
]
sage: victor_miller_basis(2, 6)
[]
sage: victor_miller_basis(4, 6)
[
1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + O(q^6)
]
sage: victor_miller_basis(6, 6, var='w')
[
1 - 504*w - 16632*w^2 - 122976*w^3 - 532728*w^4 - 1575504*w^5 + O(w^6)
]
sage: victor_miller_basis(6, 6)
[
1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 + O(q^6)
]
sage: victor_miller_basis(12, 6)
[
1 + 196560*q^2 + 16773120*q^3 + 398034000*q^4 + 4629381120*q^5 + O(q^6),
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)
]
sage: victor_miller_basis(12, 6, cusp_only=True)
[
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)
]
sage: victor_miller_basis(24, 6, cusp_only=True)
[
q + 195660*q^3 + 12080128*q^4 + 44656110*q^5 + O(q^6),
q^2 - 48*q^3 + 1080*q^4 - 15040*q^5 + O(q^6)
]
sage: victor_miller_basis(24, 6)
[
1 + 52416000*q^3 + 39007332000*q^4 + 6609020221440*q^5 + O(q^6),
q + 195660*q^3 + 12080128*q^4 + 44656110*q^5 + O(q^6),
q^2 - 48*q^3 + 1080*q^4 - 15040*q^5 + O(q^6)
]
sage: victor_miller_basis(32, 6)
[
1 + 2611200*q^3 + 19524758400*q^4 + 19715347537920*q^5 + O(q^6),
q + 50220*q^3 + 87866368*q^4 + 18647219790*q^5 + O(q^6),
q^2 + 432*q^3 + 39960*q^4 - 1418560*q^5 + O(q^6)
]
sage: victor_miller_basis(40,200)[1:] == victor_miller_basis(40,200,cusp_only=True)
True
sage: victor_miller_basis(200,40)[1:] == victor_miller_basis(200,40,cusp_only=True)
True
AUTHORS:
- William Stein, Craig Citro: original code
- Martin Raum (2009-08-02): use FLINT for polynomial arithmetic (instead of NTL)
"""
k = Integer(k)
if k % 2 == 1 or k == 2:
return Sequence([])
elif k < 0:
raise ValueError("k must be non-negative")
elif k == 0:
return Sequence([PowerSeriesRing(ZZ, var)(1).add_bigoh(prec)], cr=True)
e = k.mod(12)
if e == 2: e += 12
n = (k - e) // 12
if n == 0 and cusp_only:
return Sequence([])
# If prec is less than or equal to the dimension of the space of
# cusp forms, which is just n, then we know the answer, and we
# simply return it.
if prec <= n:
q = PowerSeriesRing(ZZ, var).gen(0)
err = bigO(q**prec)
ls = [0] * (n + 1)
if not cusp_only:
ls[0] = 1 + err
for i in range(1, prec):
ls[i] = q**i + err
for i in range(prec, n + 1):
ls[i] = err
return Sequence(ls, cr=True)
F6 = eisenstein_series_poly(6, prec)
if e == 0:
A = Fmpz_poly(1)
elif e == 4:
A = eisenstein_series_poly(4, prec)
elif e == 6:
A = F6
elif e == 8:
A = eisenstein_series_poly(8, prec)
elif e == 10:
A = eisenstein_series_poly(10, prec)
else: # e == 14
A = eisenstein_series_poly(14, prec)
if A[0] == -1:
A = -A
if n == 0:
return Sequence([PowerSeriesRing(ZZ, var)(A.list()).add_bigoh(prec)],
cr=True)
F6_squared = F6**2
F6_squared._unsafe_mutate_truncate(prec)
D = _delta_poly(prec)
Fprod = F6_squared
Dprod = D
if cusp_only:
ls = [Fmpz_poly(0)] + [A] * n
else:
ls = [A] * (n + 1)
for i in xrange(1, n + 1):
ls[n - i] *= Fprod
ls[i] *= Dprod
ls[n - i]._unsafe_mutate_truncate(prec)
ls[i]._unsafe_mutate_truncate(prec)
Fprod *= F6_squared
Dprod *= D
Fprod._unsafe_mutate_truncate(prec)
Dprod._unsafe_mutate_truncate(prec)
P = PowerSeriesRing(ZZ, var)
if cusp_only:
for i in xrange(1, n + 1):
for j in xrange(1, i):
ls[j] = ls[j] - ls[j][i] * ls[i]
return Sequence([P(l.list()).add_bigoh(prec) for l in ls[1:]], cr=True)
else:
for i in xrange(1, n + 1):
for j in xrange(i):
ls[j] = ls[j] - ls[j][i] * ls[i]
return Sequence([P(l.list()).add_bigoh(prec) for l in ls], cr=True)
