def eisenstein_series_lseries(weight, prec=53,
max_imaginary_part=0,
max_asymp_coeffs=40):
r"""
Return the L-series of the weight `2k` Eisenstein series
on `\mathrm{SL}_2(\ZZ)`.
This actually returns an interface to Tim Dokchitser's program
for computing with the L-series of the Eisenstein series
INPUT:
- ``weight`` - even integer
- ``prec`` - integer (bits precision)
- ``max_imaginary_part`` - real number
- ``max_asymp_coeffs`` - integer
OUTPUT:
The L-series of the Eisenstein series.
EXAMPLES:
We compute with the L-series of `E_{16}` and then `E_{20}`::
sage: L = eisenstein_series_lseries(16)
sage: L(1)
-0.291657724743874
sage: L = eisenstein_series_lseries(20)
sage: L(2)
-5.02355351645998
Now with higher precision::
sage: L = eisenstein_series_lseries(20, prec=200)
sage: L(2)
-5.0235535164599797471968418348135050804419155747868718371029
"""
f = eisenstein_series_qexp(weight,prec)
from sage.lfunctions.all import Dokchitser
from sage.symbolic.constants import pi
key = (prec, max_imaginary_part, max_asymp_coeffs)
j = weight
L = Dokchitser(conductor = 1,
gammaV = [0,1],
weight = j,
eps = (-1)**Integer(j/2),
poles = [j],
# Using a string for residues is a hack but it works well
# since this will make PARI/GP compute sqrt(pi) with the
# right precision.
residues = '[sqrt(Pi)*(%s)]'%((-1)**Integer(j/2)*bernoulli(j)/j),
prec = prec)
s = 'coeff = %s;'%f.list()
L.init_coeffs('coeff[k+1]',pari_precode = s,
max_imaginary_part=max_imaginary_part,
max_asymp_coeffs=max_asymp_coeffs)
L.check_functional_equation()
L.rename('L-series associated to the weight %s Eisenstein series %s on SL_2(Z)'%(j,f))
return L
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 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 eisenstein_series_lseries(weight,
prec=53,
max_imaginary_part=0,
max_asymp_coeffs=40):
r"""
Return the L-series of the weight `2k` Eisenstein series
on `\mathrm{SL}_2(\ZZ)`.
This actually returns an interface to Tim Dokchitser's program
for computing with the L-series of the Eisenstein series
INPUT:
- ``weight`` - even integer
- ``prec`` - integer (bits precision)
- ``max_imaginary_part`` - real number
- ``max_asymp_coeffs`` - integer
OUTPUT:
The L-series of the Eisenstein series.
EXAMPLES:
We compute with the L-series of `E_{16}` and then `E_{20}`::
sage: L = eisenstein_series_lseries(16)
sage: L(1)
-0.291657724743874
sage: L = eisenstein_series_lseries(20)
sage: L(2)
-5.02355351645998
Now with higher precision::
sage: L = eisenstein_series_lseries(20, prec=200)
sage: L(2)
-5.0235535164599797471968418348135050804419155747868718371029
"""
f = eisenstein_series_qexp(weight, prec)
from sage.lfunctions.all import Dokchitser
from sage.symbolic.constants import pi
key = (prec, max_imaginary_part, max_asymp_coeffs)
j = weight
L = Dokchitser(
conductor=1,
gammaV=[0, 1],
weight=j,
eps=(-1)**Integer(j / 2),
poles=[j],
# Using a string for residues is a hack but it works well
# since this will make PARI/GP compute sqrt(pi) with the
# right precision.
residues='[sqrt(Pi)*(%s)]' % ((-1)**Integer(j / 2) * bernoulli(j) / j),
prec=prec)
s = 'coeff = %s;' % f.list()
L.init_coeffs('coeff[k+1]',
pari_precode=s,
max_imaginary_part=max_imaginary_part,
max_asymp_coeffs=max_asymp_coeffs)
L.check_functional_equation()
L.rename(
'L-series associated to the weight %s Eisenstein series %s on SL_2(Z)'
% (j, f))
return L
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)
