def eisenstein_series_qexp(k, prec = 10, K=QQ, var='q', normalization='linear'):
r"""
Return the `q`-expansion of the normalized weight `k` Eisenstein series on
`{\rm SL}_2(\ZZ)` to precision prec in the ring `K`. Three normalizations
are available, depending on the parameter ``normalization``; the default
normalization is the one for which the linear coefficient is 1.
INPUT:
- ``k`` - an even positive integer
- ``prec`` - (default: 10) a nonnegative integer
- ``K`` - (default: `\QQ`) a ring
- ``var`` - (default: ``'q'``) variable name to use for q-expansion
- ``normalization`` - (default: ``'linear'``) normalization to use. If this
is ``'linear'``, then the series will be normalized so that the linear
term is 1. If it is ``'constant'``, the series will be normalized to have
constant term 1. If it is ``'integral'``, then the series will be
normalized to have integer coefficients and no common factor, and linear
term that is positive. Note that ``'integral'`` will work over arbitrary
base rings, while ``'linear'`` or ``'constant'`` will fail if the
denominator (resp. numerator) of `B_k / 2k` is invertible.
ALGORITHM:
We know `E_k = \text{constant} + \sum_n \sigma_{k-1}(n) q^n`. So we
compute all the `\sigma_{k-1}(n)` simultaneously, using the fact that
`\sigma` is multiplicative.
EXAMPLES::
sage: eisenstein_series_qexp(2,5)
-1/24 + q + 3*q^2 + 4*q^3 + 7*q^4 + O(q^5)
sage: eisenstein_series_qexp(2,0)
O(q^0)
sage: eisenstein_series_qexp(2,5,GF(7))
2 + q + 3*q^2 + 4*q^3 + O(q^5)
sage: eisenstein_series_qexp(2,5,GF(7),var='T')
2 + T + 3*T^2 + 4*T^3 + O(T^5)
We illustrate the use of the ``normalization`` parameter::
sage: eisenstein_series_qexp(12, 5, normalization='integral')
691 + 65520*q + 134250480*q^2 + 11606736960*q^3 + 274945048560*q^4 + O(q^5)
sage: eisenstein_series_qexp(12, 5, normalization='constant')
1 + 65520/691*q + 134250480/691*q^2 + 11606736960/691*q^3 + 274945048560/691*q^4 + O(q^5)
sage: eisenstein_series_qexp(12, 5, normalization='linear')
691/65520 + q + 2049*q^2 + 177148*q^3 + 4196353*q^4 + O(q^5)
sage: eisenstein_series_qexp(12, 50, K=GF(13), normalization="constant")
1 + O(q^50)
TESTS:
Test that :trac:`5102` is fixed::
sage: eisenstein_series_qexp(10, 30, GF(17))
15 + q + 3*q^2 + 15*q^3 + 7*q^4 + 13*q^5 + 11*q^6 + 11*q^7 + 15*q^8 + 7*q^9 + 5*q^10 + 7*q^11 + 3*q^12 + 14*q^13 + 16*q^14 + 8*q^15 + 14*q^16 + q^17 + 4*q^18 + 3*q^19 + 6*q^20 + 12*q^21 + 4*q^22 + 12*q^23 + 4*q^24 + 4*q^25 + 8*q^26 + 14*q^27 + 9*q^28 + 6*q^29 + O(q^30)
This shows that the bug reported at :trac:`8291` is fixed::
sage: eisenstein_series_qexp(26, 10, GF(13))
7 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + 12*q^6 + 8*q^7 + 2*q^8 + O(q^10)
We check that the function behaves properly over finite-characteristic base rings::
sage: eisenstein_series_qexp(12, 5, K = Zmod(691), normalization="integral")
566*q + 236*q^2 + 286*q^3 + 194*q^4 + O(q^5)
sage: eisenstein_series_qexp(12, 5, K = Zmod(691), normalization="constant")
Traceback (most recent call last):
...
ValueError: The numerator of -B_k/(2*k) (=691) must be invertible in the ring Ring of integers modulo 691
sage: eisenstein_series_qexp(12, 5, K = Zmod(691), normalization="linear")
q + 667*q^2 + 252*q^3 + 601*q^4 + O(q^5)
sage: eisenstein_series_qexp(12, 5, K = Zmod(2), normalization="integral")
1 + O(q^5)
sage: eisenstein_series_qexp(12, 5, K = Zmod(2), normalization="constant")
1 + O(q^5)
sage: eisenstein_series_qexp(12, 5, K = Zmod(2), normalization="linear")
Traceback (most recent call last):
...
ValueError: The denominator of -B_k/(2*k) (=65520) must be invertible in the ring Ring of integers modulo 2
AUTHORS:
- William Stein: original implementation
- Craig Citro (2007-06-01): rewrote for massive speedup
- Martin Raum (2009-08-02): port to cython for speedup
- David Loeffler (2010-04-07): work around an integer overflow when `k` is large
- David Loeffler (2012-03-15): add options for alternative normalizations
(motivated by :trac:`12043`)
"""
## we use this to prevent computation if it would fail anyway.
if k <= 0 or k % 2 == 1 :
raise ValueError("k must be positive and even")
a0 = - bernoulli(k) / (2*k)
if normalization == 'linear':
a0den = a0.denominator()
try:
a0fac = K(1/a0den)
except ZeroDivisionError:
raise ValueError("The denominator of -B_k/(2*k) (=%s) must be invertible in the ring %s"%(a0den, K))
elif normalization == 'constant':
a0num = a0.numerator()
try:
a0fac = K(1/a0num)
except ZeroDivisionError:
raise ValueError("The numerator of -B_k/(2*k) (=%s) must be invertible in the ring %s"%(a0num, K))
elif normalization == 'integral':
a0fac = None
else:
raise ValueError("Normalization (=%s) must be one of 'linear', 'constant', 'integral'" % normalization)
R = PowerSeriesRing(K, var)
if K == QQ and normalization == 'linear':
ls = Ek_ZZ(k, prec)
# The following is *dramatically* faster than doing the more natural
# "R(ls)" would be:
E = ZZ[var](ls, prec=prec, check=False).change_ring(QQ)
if len(ls)>0:
E._unsafe_mutate(0, a0)
return R(E, prec)
# The following is an older slower alternative to the above three lines:
#return a0fac*R(eisenstein_series_poly(k, prec).list(), prec=prec, check=False)
else:
# This used to work with check=False, but that can only be regarded as
# an improbable lucky miracle. Enabling checking is a noticeable speed
# regression; the morally right fix would be to expose FLINT's
# fmpz_poly_to_nmod_poly command (at least for word-sized N).
if a0fac is not None:
return a0fac*R(eisenstein_series_poly(k, prec).list(), prec=prec, check=True)
else:
return R(eisenstein_series_poly(k, prec).list(), prec=prec, check=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(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 eisenstein_series_qexp(k, prec = 10, K=QQ, var='q') :
r"""
Return the `q`-expansion of the normalized weight `k` Eisenstein series on
`{\rm SL}_2(\ZZ)` to precision prec in the ring `K`. (The normalization
chosen here is the one that forces the coefficient of `q` to be 1.)
INPUT:
- ``k`` - an even positive integer
- ``prec`` - (default: 10) a nonnegative integer
- ``K`` - (default: `\QQ`) a ring in which the denominator of `B_k / 2k` is invertible
- ``var`` - (default: 'q') variable name to use for q-expansion
ALGORITHM:
We know `E_k = \text{constant} + \sum_n \sigma_{k-1}(n) q^n`. So we
compute all the `\sigma_{k-1}(n)` simultaneously, using the fact that
`\sigma` is multiplicative.
EXAMPLES::
sage: eisenstein_series_qexp(2,5)
-1/24 + q + 3*q^2 + 4*q^3 + 7*q^4 + O(q^5)
sage: eisenstein_series_qexp(2,0)
O(q^0)
sage: eisenstein_series_qexp(2,5,GF(7))
2 + q + 3*q^2 + 4*q^3 + O(q^5)
sage: eisenstein_series_qexp(2,5,GF(7),var='T')
2 + T + 3*T^2 + 4*T^3 + O(T^5)
sage: eisenstein_series_qexp(10, 30, GF(17))
15 + q + 3*q^2 + 15*q^3 + 7*q^4 + 13*q^5 + 11*q^6 + 11*q^7 + 15*q^8 + 7*q^9 + 5*q^10 + 7*q^11 + 3*q^12 + 14*q^13 + 16*q^14 + 8*q^15 + 14*q^16 + q^17 + 4*q^18 + 3*q^19 + 6*q^20 + 12*q^21 + 4*q^22 + 12*q^23 + 4*q^24 + 4*q^25 + 8*q^26 + 14*q^27 + 9*q^28 + 6*q^29 + O(q^30)
TESTS:
This shows that the bug reported at trac 8291 is fixed::
sage: eisenstein_series_qexp(26, 10, GF(13))
7 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + 12*q^6 + 8*q^7 + 2*q^8 + O(q^10)
AUTHORS:
- William Stein: original implementation
- Craig Citro (2007-06-01): rewrote for massive speedup
- Martin Raum (2009-08-02): port to cython for speedup
- David Loeffler (2010-04-07): work around an integer overflow when k is large
"""
## we use this to prevent computation if it would fail anyway.
if k <= 0 or k % 2 == 1 :
raise ValueError, "k must be positive and even"
a0 = - bernoulli(k) / (2*k)
a0den = a0.denominator()
try:
a0fac = K(1/a0den)
except ZeroDivisionError:
raise ValueError, "The denominator of -B_k/(2*k) (=%s) must be invertible in the ring %s"%(a0den, K)
R = PowerSeriesRing(K, var)
if K == QQ:
ls = Ek_ZZ(k, prec)
# The following is *dramatically* faster than doing the more natural
# "R(ls)" would be:
E = ZZ[var](ls, prec=prec, check=False).change_ring(QQ)
if len(ls)>0:
E._unsafe_mutate(0, a0)
return R(E, prec)
# The following is an older slower alternative to the above three lines:
#return a0fac*R(eisenstein_series_poly(k, prec).list(), prec=prec, check=False)
else:
# this is a temporary fix due to a change in the
# polynomial constructor over finite fields; this
# is a notable speed regression, to be fixed soon.
return a0fac*R(eisenstein_series_poly(k, prec).list(), prec=prec, check=True)
def eisenstein_series_qexp(k, prec=10, K=QQ, var='q', normalization='linear'):
r"""
Return the `q`-expansion of the normalized weight `k` Eisenstein series on
`{\rm SL}_2(\ZZ)` to precision prec in the ring `K`. Three normalizations
are available, depending on the parameter ``normalization``; the default
normalization is the one for which the linear coefficient is 1.
INPUT:
- ``k`` - an even positive integer
- ``prec`` - (default: 10) a nonnegative integer
- ``K`` - (default: `\QQ`) a ring
- ``var`` - (default: ``'q'``) variable name to use for q-expansion
- ``normalization`` - (default: ``'linear'``) normalization to use. If this
is ``'linear'``, then the series will be normalized so that the linear
term is 1. If it is ``'constant'``, the series will be normalized to have
constant term 1. If it is ``'integral'``, then the series will be
normalized to have integer coefficients and no common factor, and linear
term that is positive. Note that ``'integral'`` will work over arbitrary
base rings, while ``'linear'`` or ``'constant'`` will fail if the
denominator (resp. numerator) of `B_k / 2k` is invertible.
ALGORITHM:
We know `E_k = \text{constant} + \sum_n \sigma_{k-1}(n) q^n`. So we
compute all the `\sigma_{k-1}(n)` simultaneously, using the fact that
`\sigma` is multiplicative.
EXAMPLES::
sage: eisenstein_series_qexp(2,5)
-1/24 + q + 3*q^2 + 4*q^3 + 7*q^4 + O(q^5)
sage: eisenstein_series_qexp(2,0)
O(q^0)
sage: eisenstein_series_qexp(2,5,GF(7))
2 + q + 3*q^2 + 4*q^3 + O(q^5)
sage: eisenstein_series_qexp(2,5,GF(7),var='T')
2 + T + 3*T^2 + 4*T^3 + O(T^5)
We illustrate the use of the ``normalization`` parameter::
sage: eisenstein_series_qexp(12, 5, normalization='integral')
691 + 65520*q + 134250480*q^2 + 11606736960*q^3 + 274945048560*q^4 + O(q^5)
sage: eisenstein_series_qexp(12, 5, normalization='constant')
1 + 65520/691*q + 134250480/691*q^2 + 11606736960/691*q^3 + 274945048560/691*q^4 + O(q^5)
sage: eisenstein_series_qexp(12, 5, normalization='linear')
691/65520 + q + 2049*q^2 + 177148*q^3 + 4196353*q^4 + O(q^5)
sage: eisenstein_series_qexp(12, 50, K=GF(13), normalization="constant")
1 + O(q^50)
TESTS:
Test that :trac:`5102` is fixed::
sage: eisenstein_series_qexp(10, 30, GF(17))
15 + q + 3*q^2 + 15*q^3 + 7*q^4 + 13*q^5 + 11*q^6 + 11*q^7 + 15*q^8 + 7*q^9 + 5*q^10 + 7*q^11 + 3*q^12 + 14*q^13 + 16*q^14 + 8*q^15 + 14*q^16 + q^17 + 4*q^18 + 3*q^19 + 6*q^20 + 12*q^21 + 4*q^22 + 12*q^23 + 4*q^24 + 4*q^25 + 8*q^26 + 14*q^27 + 9*q^28 + 6*q^29 + O(q^30)
This shows that the bug reported at :trac:`8291` is fixed::
sage: eisenstein_series_qexp(26, 10, GF(13))
7 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + 12*q^6 + 8*q^7 + 2*q^8 + O(q^10)
We check that the function behaves properly over finite-characteristic base rings::
sage: eisenstein_series_qexp(12, 5, K = Zmod(691), normalization="integral")
566*q + 236*q^2 + 286*q^3 + 194*q^4 + O(q^5)
sage: eisenstein_series_qexp(12, 5, K = Zmod(691), normalization="constant")
Traceback (most recent call last):
...
ValueError: The numerator of -B_k/(2*k) (=691) must be invertible in the ring Ring of integers modulo 691
sage: eisenstein_series_qexp(12, 5, K = Zmod(691), normalization="linear")
q + 667*q^2 + 252*q^3 + 601*q^4 + O(q^5)
sage: eisenstein_series_qexp(12, 5, K = Zmod(2), normalization="integral")
1 + O(q^5)
sage: eisenstein_series_qexp(12, 5, K = Zmod(2), normalization="constant")
1 + O(q^5)
sage: eisenstein_series_qexp(12, 5, K = Zmod(2), normalization="linear")
Traceback (most recent call last):
...
ValueError: The denominator of -B_k/(2*k) (=65520) must be invertible in the ring Ring of integers modulo 2
AUTHORS:
- William Stein: original implementation
- Craig Citro (2007-06-01): rewrote for massive speedup
- Martin Raum (2009-08-02): port to cython for speedup
- David Loeffler (2010-04-07): work around an integer overflow when `k` is large
- David Loeffler (2012-03-15): add options for alternative normalizations
(motivated by :trac:`12043`)
"""
## we use this to prevent computation if it would fail anyway.
if k <= 0 or k % 2 == 1:
raise ValueError, "k must be positive and even"
a0 = -bernoulli(k) / (2 * k)
if normalization == 'linear':
a0den = a0.denominator()
try:
a0fac = K(1 / a0den)
except ZeroDivisionError:
raise ValueError, "The denominator of -B_k/(2*k) (=%s) must be invertible in the ring %s" % (
a0den, K)
elif normalization == 'constant':
a0num = a0.numerator()
try:
a0fac = K(1 / a0num)
except ZeroDivisionError:
raise ValueError, "The numerator of -B_k/(2*k) (=%s) must be invertible in the ring %s" % (
a0num, K)
elif normalization == 'integral':
a0fac = None
else:
raise ValueError, "Normalization (=%s) must be one of 'linear', 'constant', 'integral'" % normalization
R = PowerSeriesRing(K, var)
if K == QQ and normalization == 'linear':
ls = Ek_ZZ(k, prec)
# The following is *dramatically* faster than doing the more natural
# "R(ls)" would be:
E = ZZ[var](ls, prec=prec, check=False).change_ring(QQ)
if len(ls) > 0:
E._unsafe_mutate(0, a0)
return R(E, prec)
# The following is an older slower alternative to the above three lines:
#return a0fac*R(eisenstein_series_poly(k, prec).list(), prec=prec, check=False)
else:
# This used to work with check=False, but that can only be regarded as
# an improbable lucky miracle. Enabling checking is a noticeable speed
# regression; the morally right fix would be to expose FLINT's
# fmpz_poly_to_nmod_poly command (at least for word-sized N).
if a0fac is not None:
return a0fac * R(
eisenstein_series_poly(k, prec).list(), prec=prec, check=True)
else:
return R(eisenstein_series_poly(k, prec).list(),
prec=prec,
check=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(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)
