def ap(self, p):
"""
Return a list of the eigenvalues of the Hecke operator `T_p`
on all the computed eigenforms. The eigenvalues match up
between one prime and the next.
INPUT:
- ``p`` - integer, a prime number
OUTPUT:
- ``list`` - a list of double precision complex numbers
EXAMPLES::
sage: n = numerical_eigenforms(11,4)
sage: n.ap(2) # random order
[9.0, 9.0, 2.73205080757, -0.732050807569]
sage: n.ap(3) # random order
[28.0, 28.0, -7.92820323028, 5.92820323028]
sage: m = n.modular_symbols()
sage: x = polygen(QQ, 'x')
sage: m.T(2).charpoly('x').factor()
(x - 9)^2 * (x^2 - 2*x - 2)
sage: m.T(3).charpoly('x').factor()
(x - 28)^2 * (x^2 + 2*x - 47)
"""
p = Integer(p)
if not p.is_prime():
raise ValueError("p must be a prime")
return Sequence(self.eigenvalues([p])[0], immutable=True)
def __init__(self, abvar, p):
"""
Create a `p`-adic `L`-series.
EXAMPLES::
sage: J0(37)[0].padic_lseries(389)
389-adic L-series attached to Simple abelian subvariety 37a(1,37) of dimension 1 of J0(37)
"""
Lseries.__init__(self, abvar)
p = Integer(p)
if not p.is_prime():
raise ValueError("p (=%s) must be prime"%p)
self.__p = p
def __init__(self, abvar, p):
"""
Create a `p`-adic `L`-series.
EXAMPLES::
sage: J0(37)[0].padic_lseries(389)
389-adic L-series attached to Simple abelian subvariety 37a(1,37) of dimension 1 of J0(37)
"""
Lseries.__init__(self, abvar)
p = Integer(p)
if not p.is_prime():
raise ValueError("p (=%s) must be prime" % p)
self.__p = p
def ap(self, p):
"""
Return a list of the eigenvalues of the Hecke operator `T_p`
on all the computed eigenforms. The eigenvalues match up
between one prime and the next.
INPUT:
- ``p`` - integer, a prime number
OUTPUT:
- ``list`` - a list of double precision complex numbers
EXAMPLES::
sage: n = numerical_eigenforms(11,4)
sage: n.ap(2) # random order
[9.0, 9.0, 2.73205080757, -0.732050807569]
sage: n.ap(3) # random order
[28.0, 28.0, -7.92820323028, 5.92820323028]
sage: m = n.modular_symbols()
sage: x = polygen(QQ, 'x')
sage: m.T(2).charpoly('x').factor()
(x - 9)^2 * (x^2 - 2*x - 2)
sage: m.T(3).charpoly('x').factor()
(x - 28)^2 * (x^2 + 2*x - 47)
"""
p = Integer(p)
if not p.is_prime():
raise ValueError("p must be a prime")
try:
return self._ap[p]
except AttributeError:
self._ap = {}
except KeyError:
pass
a = Sequence(self.eigenvalues([p])[0], immutable=True)
self._ap[p] = a
return a
def check_prime(K,P):
r"""
Function to check that `P` determines a prime of `K`, and return that ideal.
INPUT:
- ``K`` -- a number field (including `\QQ`).
- ``P`` -- an element of ``K`` or a (fractional) ideal of ``K``.
OUTPUT:
- If ``K`` is `\QQ`: the prime integer equal to or which generates `P`.
- If ``K`` is not `\QQ`: the prime ideal equal to or generated by `P`.
.. note::
If `P` is not a prime and does not generate a prime, a TypeError is raised.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.ell_local_data import check_prime
sage: check_prime(QQ,3)
3
sage: check_prime(QQ,ZZ.ideal(31))
31
sage: K.<a>=NumberField(x^2-5)
sage: check_prime(K,a)
Fractional ideal (a)
sage: check_prime(K,a+1)
Fractional ideal (a + 1)
sage: [check_prime(K,P) for P in K.primes_above(31)]
[Fractional ideal (5/2*a + 1/2), Fractional ideal (5/2*a - 1/2)]
"""
if K is QQ:
if isinstance(P, (int,long,Integer)):
P = Integer(P)
if P.is_prime():
return P
else:
raise TypeError("%s is not prime"%P)
else:
if is_Ideal(P) and P.base_ring() is ZZ and P.is_prime():
return P.gen()
raise TypeError("%s is not a prime ideal of %s"%(P,ZZ))
if not is_NumberField(K):
raise TypeError("%s is not a number field"%K)
if is_NumberFieldFractionalIdeal(P):
if P.is_prime():
return P
else:
raise TypeError("%s is not a prime ideal of %s"%(P,K))
if is_NumberFieldElement(P):
if P in K:
P = K.ideal(P)
else:
raise TypeError("%s is not an element of %s"%(P,K))
if P.is_prime():
return P
else:
raise TypeError("%s is not a prime ideal of %s"%(P,K))
raise TypeError("%s is not a valid prime of %s"%(P,K))
def check_prime(K, P):
r"""
Function to check that `P` determines a prime of `K`, and return that ideal.
INPUT:
- ``K`` -- a number field (including `\QQ`).
- ``P`` -- an element of ``K`` or a (fractional) ideal of ``K``.
OUTPUT:
- If ``K`` is `\QQ`: the prime integer equal to or which generates `P`.
- If ``K`` is not `\QQ`: the prime ideal equal to or generated by `P`.
.. NOTE::
If `P` is not a prime and does not generate a prime, a ``TypeError``
is raised.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.ell_local_data import check_prime
sage: check_prime(QQ,3)
3
sage: check_prime(QQ,QQ(3))
3
sage: check_prime(QQ,ZZ.ideal(31))
31
sage: K.<a> = NumberField(x^2-5)
sage: check_prime(K,a)
Fractional ideal (a)
sage: check_prime(K,a+1)
Fractional ideal (a + 1)
sage: [check_prime(K,P) for P in K.primes_above(31)]
[Fractional ideal (5/2*a + 1/2), Fractional ideal (5/2*a - 1/2)]
sage: L.<b> = NumberField(x^2+3)
sage: check_prime(K, L.ideal(5))
Traceback (most recent call last):
...
TypeError: The ideal Fractional ideal (5) is not a prime ideal of Number Field in a with defining polynomial x^2 - 5
sage: check_prime(K, L.ideal(b))
Traceback (most recent call last):
...
TypeError: No compatible natural embeddings found for Number Field in a with defining polynomial x^2 - 5 and Number Field in b with defining polynomial x^2 + 3
"""
if K is QQ:
if P in ZZ or isinstance(P, integer_types + (Integer,)):
P = Integer(P)
if P.is_prime():
return P
else:
raise TypeError("The element %s is not prime" % (P,))
elif P in QQ:
raise TypeError("The element %s is not prime" % (P,))
elif is_Ideal(P) and P.base_ring() is ZZ:
if P.is_prime():
return P.gen()
else:
raise TypeError("The ideal %s is not a prime ideal of %s" % (P, ZZ))
else:
raise TypeError("%s is neither an element of QQ or an ideal of %s" % (P, ZZ))
if not is_NumberField(K):
raise TypeError("%s is not a number field" % (K,))
if is_NumberFieldFractionalIdeal(P) or P in K:
# if P is an ideal, making sure it is an fractional ideal of K
P = K.fractional_ideal(P)
if P.is_prime():
return P
else:
raise TypeError("The ideal %s is not a prime ideal of %s" % (P, K))
raise TypeError("%s is not a valid prime of %s" % (P, K))
def check_prime(K,P):
r"""
Function to check that `P` determines a prime of `K`, and return that ideal.
INPUT:
- ``K`` -- a number field (including `\QQ`).
- ``P`` -- an element of ``K`` or a (fractional) ideal of ``K``.
OUTPUT:
- If ``K`` is `\QQ`: the prime integer equal to or which generates `P`.
- If ``K`` is not `\QQ`: the prime ideal equal to or generated by `P`.
.. note::
If `P` is not a prime and does not generate a prime, a TypeError is raised.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.ell_local_data import check_prime
sage: check_prime(QQ,3)
3
sage: check_prime(QQ,ZZ.ideal(31))
31
sage: K.<a>=NumberField(x^2-5)
sage: check_prime(K,a)
Fractional ideal (a)
sage: check_prime(K,a+1)
Fractional ideal (a + 1)
sage: [check_prime(K,P) for P in K.primes_above(31)]
[Fractional ideal (5/2*a + 1/2), Fractional ideal (5/2*a - 1/2)]
"""
if K is QQ:
if isinstance(P, (int,long,Integer)):
P = Integer(P)
if P.is_prime():
return P
else:
raise TypeError("%s is not prime"%P)
else:
if is_Ideal(P) and P.base_ring() is ZZ and P.is_prime():
return P.gen()
raise TypeError("%s is not a prime ideal of %s"%(P,ZZ))
if not is_NumberField(K):
raise TypeError("%s is not a number field"%K)
if is_NumberFieldFractionalIdeal(P):
if P.is_prime():
return P
else:
raise TypeError("%s is not a prime ideal of %s"%(P,K))
if is_NumberFieldElement(P):
if P in K:
P = K.ideal(P)
else:
raise TypeError("%s is not an element of %s"%(P,K))
if P.is_prime():
return P
else:
raise TypeError("%s is not a prime ideal of %s"%(P,K))
raise TypeError("%s is not a valid prime of %s"%(P,K))
def hecke_operator_on_qexp(f, n, k, eps = None,
prec=None, check=True, _return_list=False):
r"""
Given the `q`-expansion `f` of a modular form with character
`\varepsilon`, this function computes the image of `f` under the
Hecke operator `T_{n,k}` of weight `k`.
EXAMPLES::
sage: M = ModularForms(1,12)
sage: hecke_operator_on_qexp(M.basis()[0], 3, 12)
252*q - 6048*q^2 + 63504*q^3 - 370944*q^4 + O(q^5)
sage: hecke_operator_on_qexp(M.basis()[0], 1, 12, prec=7)
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 + O(q^7)
sage: hecke_operator_on_qexp(M.basis()[0], 1, 12)
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + 84480*q^8 - 113643*q^9 - 115920*q^10 + 534612*q^11 - 370944*q^12 - 577738*q^13 + O(q^14)
sage: M.prec(20)
20
sage: hecke_operator_on_qexp(M.basis()[0], 3, 12)
252*q - 6048*q^2 + 63504*q^3 - 370944*q^4 + 1217160*q^5 - 1524096*q^6 + O(q^7)
sage: hecke_operator_on_qexp(M.basis()[0], 1, 12)
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + 84480*q^8 - 113643*q^9 - 115920*q^10 + 534612*q^11 - 370944*q^12 - 577738*q^13 + 401856*q^14 + 1217160*q^15 + 987136*q^16 - 6905934*q^17 + 2727432*q^18 + 10661420*q^19 - 7109760*q^20 + O(q^21)
sage: (hecke_operator_on_qexp(M.basis()[0], 1, 12)*252).add_bigoh(7)
252*q - 6048*q^2 + 63504*q^3 - 370944*q^4 + 1217160*q^5 - 1524096*q^6 + O(q^7)
sage: hecke_operator_on_qexp(M.basis()[0], 6, 12)
-6048*q + 145152*q^2 - 1524096*q^3 + O(q^4)
An example on a formal power series::
sage: R.<q> = QQ[[]]
sage: f = q + q^2 + q^3 + q^7 + O(q^8)
sage: hecke_operator_on_qexp(f, 3, 12)
q + O(q^3)
sage: hecke_operator_on_qexp(delta_qexp(24), 3, 12).prec()
8
sage: hecke_operator_on_qexp(delta_qexp(25), 3, 12).prec()
9
An example of computing `T_{p,k}` in characteristic `p`::
sage: p = 199
sage: fp = delta_qexp(prec=p^2+1, K=GF(p))
sage: tfp = hecke_operator_on_qexp(fp, p, 12)
sage: tfp == fp[p] * fp
True
sage: tf = hecke_operator_on_qexp(delta_qexp(prec=p^2+1), p, 12).change_ring(GF(p))
sage: tfp == tf
True
"""
if eps is None:
# Need to have base_ring=ZZ to work over finite fields, since
# ZZ can coerce to GF(p), but QQ can't.
eps = DirichletGroup(1, base_ring=ZZ)[0]
if check:
if not (is_PowerSeries(f) or is_ModularFormElement(f)):
raise TypeError("f (=%s) must be a power series or modular form"%f)
if not is_DirichletCharacter(eps):
raise TypeError("eps (=%s) must be a Dirichlet character"%eps)
k = Integer(k)
n = Integer(n)
v = []
if prec is None:
if is_ModularFormElement(f):
# always want at least three coefficients, but not too many, unless
# requested
pr = max(f.prec(), f.parent().prec(), (n+1)*3)
pr = min(pr, 100*(n+1))
prec = pr // n + 1
else:
prec = (f.prec() / ZZ(n)).ceil()
if prec == Infinity: prec = f.parent().default_prec() // n + 1
if f.prec() < prec:
f._compute_q_expansion(prec)
p = Integer(f.base_ring().characteristic())
if k != 1 and p.is_prime() and n.is_power_of(p):
# if computing T_{p^a} in characteristic p, use the simpler (and faster)
# formula
v = [f[m*n] for m in range(prec)]
else:
l = k-1
for m in range(prec):
am = sum([eps(d) * d**l * f[m*n//(d*d)] for \
d in divisors(gcd(n, m)) if (m*n) % (d*d) == 0])
v.append(am)
if _return_list:
return v
if is_ModularFormElement(f):
R = f.parent()._q_expansion_ring()
else:
R = f.parent()
return R(v, prec)
def hecke_operator_on_qexp(f,
n,
k,
eps=None,
prec=None,
check=True,
_return_list=False):
r"""
Given the `q`-expansion `f` of a modular form with character
`\varepsilon`, this function computes the image of `f` under the
Hecke operator `T_{n,k}` of weight `k`.
EXAMPLES::
sage: M = ModularForms(1,12)
sage: hecke_operator_on_qexp(M.basis()[0], 3, 12)
252*q - 6048*q^2 + 63504*q^3 - 370944*q^4 + O(q^5)
sage: hecke_operator_on_qexp(M.basis()[0], 1, 12, prec=7)
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 + O(q^7)
sage: hecke_operator_on_qexp(M.basis()[0], 1, 12)
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + 84480*q^8 - 113643*q^9 - 115920*q^10 + 534612*q^11 - 370944*q^12 - 577738*q^13 + O(q^14)
sage: M.prec(20)
20
sage: hecke_operator_on_qexp(M.basis()[0], 3, 12)
252*q - 6048*q^2 + 63504*q^3 - 370944*q^4 + 1217160*q^5 - 1524096*q^6 + O(q^7)
sage: hecke_operator_on_qexp(M.basis()[0], 1, 12)
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + 84480*q^8 - 113643*q^9 - 115920*q^10 + 534612*q^11 - 370944*q^12 - 577738*q^13 + 401856*q^14 + 1217160*q^15 + 987136*q^16 - 6905934*q^17 + 2727432*q^18 + 10661420*q^19 - 7109760*q^20 + O(q^21)
sage: (hecke_operator_on_qexp(M.basis()[0], 1, 12)*252).add_bigoh(7)
252*q - 6048*q^2 + 63504*q^3 - 370944*q^4 + 1217160*q^5 - 1524096*q^6 + O(q^7)
sage: hecke_operator_on_qexp(M.basis()[0], 6, 12)
-6048*q + 145152*q^2 - 1524096*q^3 + O(q^4)
An example on a formal power series::
sage: R.<q> = QQ[[]]
sage: f = q + q^2 + q^3 + q^7 + O(q^8)
sage: hecke_operator_on_qexp(f, 3, 12)
q + O(q^3)
sage: hecke_operator_on_qexp(delta_qexp(24), 3, 12).prec()
8
sage: hecke_operator_on_qexp(delta_qexp(25), 3, 12).prec()
9
An example of computing `T_{p,k}` in characteristic `p`::
sage: p = 199
sage: fp = delta_qexp(prec=p^2+1, K=GF(p))
sage: tfp = hecke_operator_on_qexp(fp, p, 12)
sage: tfp == fp[p] * fp
True
sage: tf = hecke_operator_on_qexp(delta_qexp(prec=p^2+1), p, 12).change_ring(GF(p))
sage: tfp == tf
True
"""
if eps is None:
# Need to have base_ring=ZZ to work over finite fields, since
# ZZ can coerce to GF(p), but QQ can't.
eps = DirichletGroup(1, base_ring=ZZ).gen(0)
if check:
if not (is_PowerSeries(f) or is_ModularFormElement(f)):
raise TypeError, "f (=%s) must be a power series or modular form" % f
if not is_DirichletCharacter(eps):
raise TypeError, "eps (=%s) must be a Dirichlet character" % eps
k = Integer(k)
n = Integer(n)
v = []
if prec is None:
if is_ModularFormElement(f):
# always want at least three coefficients, but not too many, unless
# requested
pr = max(f.prec(), f.parent().prec(), (n + 1) * 3)
pr = min(pr, 100 * (n + 1))
prec = pr // n + 1
else:
prec = (f.prec() / ZZ(n)).ceil()
if prec == Infinity: prec = f.parent().default_prec() // n + 1
if f.prec() < prec:
f._compute_q_expansion(prec)
p = Integer(f.base_ring().characteristic())
if k != 1 and p.is_prime() and n.is_power_of(p):
# if computing T_{p^a} in characteristic p, use the simpler (and faster)
# formula
v = [f[m * n] for m in range(prec)]
else:
l = k - 1
for m in range(prec):
am = sum([eps(d) * d**l * f[m*n//(d*d)] for \
d in divisors(gcd(n, m)) if (m*n) % (d*d) == 0])
v.append(am)
if _return_list:
return v
if is_ModularFormElement(f):
R = f.parent()._q_expansion_ring()
else:
R = f.parent()
return R(v, prec)
def check_prime(K,P):
r"""
Function to check that `P` determines a prime of `K`, and return that ideal.
INPUT:
- ``K`` -- a number field (including `\QQ`).
- ``P`` -- an element of ``K`` or a (fractional) ideal of ``K``.
OUTPUT:
- If ``K`` is `\QQ`: the prime integer equal to or which generates `P`.
- If ``K`` is not `\QQ`: the prime ideal equal to or generated by `P`.
.. note::
If `P` is not a prime and does not generate a prime, a TypeError is raised.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.ell_local_data import check_prime
sage: check_prime(QQ,3)
3
sage: check_prime(QQ,QQ(3))
3
sage: check_prime(QQ,ZZ.ideal(31))
31
sage: K.<a>=NumberField(x^2-5)
sage: check_prime(K,a)
Fractional ideal (a)
sage: check_prime(K,a+1)
Fractional ideal (a + 1)
sage: [check_prime(K,P) for P in K.primes_above(31)]
[Fractional ideal (5/2*a + 1/2), Fractional ideal (5/2*a - 1/2)]
sage: L.<b> = NumberField(x^2+3)
sage: check_prime(K, L.ideal(5))
Traceback (most recent call last):
..
TypeError: The ideal Fractional ideal (5) is not a prime ideal of Number Field in a with defining polynomial x^2 - 5
sage: check_prime(K, L.ideal(b))
Traceback (most recent call last):
TypeError: No compatible natural embeddings found for Number Field in a with defining polynomial x^2 - 5 and Number Field in b with defining polynomial x^2 + 3
"""
if K is QQ:
if P in ZZ or isinstance(P, integer_types + (Integer,)):
P = Integer(P)
if P.is_prime():
return P
else:
raise TypeError("The element %s is not prime" % (P,) )
elif P in QQ:
raise TypeError("The element %s is not prime" % (P,) )
elif is_Ideal(P) and P.base_ring() is ZZ:
if P.is_prime():
return P.gen()
else:
raise TypeError("The ideal %s is not a prime ideal of %s" % (P, ZZ))
else:
raise TypeError("%s is neither an element of QQ or an ideal of %s" % (P, ZZ))
if not is_NumberField(K):
raise TypeError("%s is not a number field" % (K,) )
if is_NumberFieldFractionalIdeal(P) or P in K:
# if P is an ideal, making sure it is an fractional ideal of K
P = K.fractional_ideal(P)
if P.is_prime():
return P
else:
raise TypeError("The ideal %s is not a prime ideal of %s" % (P, K))
raise TypeError("%s is not a valid prime of %s" % (P, K))
