def hecke_module_of_level(self, N):
r"""
Return the Hecke module of level N corresponding to self, which is the
domain or codomain of a degeneracy map from self. Here N must be either
a divisor or a multiple of the level of self, and a multiple of the
conductor of the character of self.
EXAMPLES::
sage: M = ModularForms(DirichletGroup(15).0, 3); M.character().conductor()
3
sage: M.hecke_module_of_level(3)
Modular Forms space of dimension 2, character [-1] and weight 3 over Rational Field
sage: M.hecke_module_of_level(5)
Traceback (most recent call last):
...
ValueError: conductor(=3) must divide M(=5)
sage: M.hecke_module_of_level(30)
Modular Forms space of dimension 16, character [-1, 1] and weight 3 over Rational Field
"""
import constructor
if N % self.level() == 0:
return constructor.ModularForms(self.character().extend(N),
self.weight(),
self.base_ring(),
prec=self.prec())
elif self.level() % N == 0:
return constructor.ModularForms(self.character().restrict(N),
self.weight(),
self.base_ring(),
prec=self.prec())
else:
raise ValueError, "N (=%s) must be a divisor or a multiple of the level of self (=%s)" % (
N, self.level())
def hecke_module_of_level(self, N):
r"""
Return the Hecke module of level N corresponding to self, which is the
domain or codomain of a degeneracy map from self. Here N must be either
a divisor or a multiple of the level of self.
EXAMPLES::
sage: ModularForms(25, 6).hecke_module_of_level(5)
Modular Forms space of dimension 3 for Congruence Subgroup Gamma0(5) of weight 6 over Rational Field
sage: ModularForms(Gamma1(4), 3).hecke_module_of_level(8)
Modular Forms space of dimension 7 for Congruence Subgroup Gamma1(8) of weight 3 over Rational Field
sage: ModularForms(Gamma1(4), 3).hecke_module_of_level(9)
Traceback (most recent call last):
...
ValueError: N (=9) must be a divisor or a multiple of the level of self (=4)
"""
if not (N % self.level() == 0 or self.level() % N == 0):
raise ValueError, "N (=%s) must be a divisor or a multiple of the level of self (=%s)" % (
N, self.level())
import constructor
return constructor.ModularForms(self.group()._new_group_from_level(N),
self.weight(),
self.base_ring(),
prec=self.prec())
def change_ring(self, base_ring):
"""
Change the base ring of this space of modular forms.
INPUT:
- ``R`` - ring
EXAMPLES::
sage: M = ModularForms(Gamma0(37),2)
sage: M.basis()
[
q + q^3 - 2*q^4 + O(q^6),
q^2 + 2*q^3 - 2*q^4 + q^5 + O(q^6),
1 + 2/3*q + 2*q^2 + 8/3*q^3 + 14/3*q^4 + 4*q^5 + O(q^6)
]
The basis after changing the base ring is the reduction modulo
`3` of an integral basis.
::
sage: M3 = M.change_ring(GF(3))
sage: M3.basis()
[
1 + q^3 + q^4 + 2*q^5 + O(q^6),
q + q^3 + q^4 + O(q^6),
q^2 + 2*q^3 + q^4 + q^5 + O(q^6)
]
"""
import constructor
M = constructor.ModularForms(self.group(),
self.weight(),
base_ring,
prec=self.prec())
return M
def half_integral_weight_modform_basis(chi, k, prec):
r"""
A basis for the space of weight `k/2` forms with character
`\chi`. The modulus of `\chi` must be divisible by
`16` and `k` must be odd and `>1`.
INPUT:
- ``chi`` - a Dirichlet character with modulus
divisible by 16
- ``k`` - an odd integer = 1
- ``prec`` - a positive integer
OUTPUT: a list of power series
.. warning::
1. This code is very slow because it requests computation of a
basis of modular forms for integral weight spaces, and that
computation is still very slow.
2. If you give an input prec that is too small, then the output
list of power series may be larger than the dimension of the
space of half-integral forms.
EXAMPLES:
We compute some half-integral weight forms of level 16\*7
::
sage: half_integral_weight_modform_basis(DirichletGroup(16*7).0^2,3,30)
[q - 2*q^2 - q^9 + 2*q^14 + 6*q^18 - 2*q^21 - 4*q^22 - q^25 + O(q^30),
q^2 - q^14 - 3*q^18 + 2*q^22 + O(q^30),
q^4 - q^8 - q^16 + q^28 + O(q^30),
q^7 - 2*q^15 + O(q^30)]
The following illustrates that choosing too low of a precision can
give an incorrect answer.
::
sage: half_integral_weight_modform_basis(DirichletGroup(16*7).0^2,3,20)
[q - 2*q^2 - q^9 + 2*q^14 + 6*q^18 + O(q^20),
q^2 - q^14 - 3*q^18 + O(q^20),
q^4 - 2*q^8 + 2*q^12 - 4*q^16 + O(q^20),
q^7 - 2*q^8 + 4*q^12 - 2*q^15 - 6*q^16 + O(q^20),
q^8 - 2*q^12 + 3*q^16 + O(q^20)]
We compute some spaces of low level and the first few possible
weights.
::
sage: half_integral_weight_modform_basis(DirichletGroup(16,QQ).1, 3, 10)
[]
sage: half_integral_weight_modform_basis(DirichletGroup(16,QQ).1, 5, 10)
[q - 2*q^3 - 2*q^5 + 4*q^7 - q^9 + O(q^10)]
sage: half_integral_weight_modform_basis(DirichletGroup(16,QQ).1, 7, 10)
[q - 2*q^2 + 4*q^3 + 4*q^4 - 10*q^5 - 16*q^7 + 19*q^9 + O(q^10),
q^2 - 2*q^3 - 2*q^4 + 4*q^5 + 4*q^7 - 8*q^9 + O(q^10),
q^3 - 2*q^5 - 2*q^7 + 4*q^9 + O(q^10)]
sage: half_integral_weight_modform_basis(DirichletGroup(16,QQ).1, 9, 10)
[q - 2*q^2 + 4*q^3 - 8*q^4 + 14*q^5 + 16*q^6 - 40*q^7 + 16*q^8 - 57*q^9 + O(q^10),
q^2 - 2*q^3 + 4*q^4 - 8*q^5 - 8*q^6 + 20*q^7 - 8*q^8 + 32*q^9 + O(q^10),
q^3 - 2*q^4 + 4*q^5 + 4*q^6 - 10*q^7 - 16*q^9 + O(q^10),
q^4 - 2*q^5 - 2*q^6 + 4*q^7 + 4*q^9 + O(q^10),
q^5 - 2*q^7 - 2*q^9 + O(q^10)]
This example once raised an error (see trac #5792).
::
sage: half_integral_weight_modform_basis(trivial_character(16),9,10)
[q - 2*q^2 + 4*q^3 - 8*q^4 + 4*q^6 - 16*q^7 + 48*q^8 - 15*q^9 + O(q^10),
q^2 - 2*q^3 + 4*q^4 - 2*q^6 + 8*q^7 - 24*q^8 + O(q^10),
q^3 - 2*q^4 - 4*q^7 + 12*q^8 + O(q^10),
q^4 - 6*q^8 + O(q^10)]
ALGORITHM: Basmaji (page 55 of his Essen thesis, "Ein Algorithmus
zur Berechnung von Hecke-Operatoren und Anwendungen auf modulare
Kurven", http://wstein.org/scans/papers/basmaji/).
Let `S = S_{k+1}(\epsilon)` be the space of cusp forms of
even integer weight `k+1` and character
`\varepsilon = \chi \psi^{(k+1)/2}`, where `\psi`
is the nontrivial mod-4 Dirichlet character. Let `U` be the
subspace of `S \times S` of elements `(a,b)` such
that `\Theta_2 a = \Theta_3 b`. Then `U` is
isomorphic to `S_{k/2}(\chi)` via the map
`(a,b) \mapsto a/\Theta_3`.
"""
if chi.modulus() % 16:
raise ValueError, "the character must have modulus divisible by 16"
if not k % 2:
raise ValueError, "k (=%s) must be odd" % k
if k < 3:
raise ValueError, "k (=%s) must be at least 3" % k
chi = chi.minimize_base_ring()
psi = chi.parent()(DirichletGroup(4, chi.base_ring()).gen())
eps = chi * psi**((k + 1) // 2)
eps = eps.minimize_base_ring()
M = constructor.ModularForms(eps, (k + 1) // 2)
C = M.cuspidal_subspace()
B = C.basis()
# This computation of S below -- of course --dominates the whole function.
#from sage.misc.all import cputime
#tm = cputime()
#print "Computing basis..."
S = [f.q_expansion(prec) for f in B]
#print "Time to compute basis", cputime(tm)
T2 = theta2_qexp(prec)
T3 = theta_qexp(prec)
n = len(S)
MS = MatrixSpace(M.base_ring(), 2 * n, prec)
A = copy(MS.zero_matrix())
for i in range(n):
T2f = T2 * S[i]
T3f = T3 * S[i]
for j in range(prec):
A[i, j] = T2f[j]
A[n + i, j] = -T3f[j]
B = A.kernel().basis()
a_vec = [sum([b[i] * S[i] for i in range(n)]) for b in B]
if len(a_vec) == 0:
return []
R = a_vec[0].parent()
t3 = R(T3)
return [R(a) / t3 for a in a_vec]
