def change_ring(self, base_ring):
"""
Return space with same defining parameters as this ambient
space of modular symbols, but defined over a different base
ring.
EXAMPLES::
sage: m = ModularForms(DirichletGroup(13).0^2,2); m
Modular Forms space of dimension 3, character [zeta6] and weight 2 over Cyclotomic Field of order 6 and degree 2
sage: m.change_ring(CyclotomicField(12))
Modular Forms space of dimension 3, character [zeta6] and weight 2 over Cyclotomic Field of order 12 and degree 4
It must be possible to change the ring of the underlying Dirichlet character::
sage: m.change_ring(QQ)
Traceback (most recent call last):
...
ValueError: cannot coerce element of order 6 into self
"""
if self.base_ring() == base_ring:
return self
return ambient_R.ModularFormsAmbient_R(self, base_ring=base_ring)
def ModularForms(group = 1,
weight = 2,
base_ring = None,
use_cache = True,
prec = defaults.DEFAULT_PRECISION):
r"""
Create an ambient space of modular forms.
INPUT:
- ``group`` - A congruence subgroup or a Dirichlet
character eps.
- ``weight`` - int, the weight, which must be an
integer = 1.
- ``base_ring`` - the base ring (ignored if group is
a Dirichlet character)
Create using the command ModularForms(group, weight, base_ring)
where group could be either a congruence subgroup or a Dirichlet
character.
EXAMPLES: First we create some spaces with trivial character::
sage: ModularForms(Gamma0(11),2).dimension()
2
sage: ModularForms(Gamma0(1),12).dimension()
2
If we give an integer N for the congruence subgroup, it defaults to
`\Gamma_0(N)`::
sage: ModularForms(1,12).dimension()
2
sage: ModularForms(11,4)
Modular Forms space of dimension 4 for Congruence Subgroup Gamma0(11) of weight 4 over Rational Field
We create some spaces for `\Gamma_1(N)`.
::
sage: ModularForms(Gamma1(13),2)
Modular Forms space of dimension 13 for Congruence Subgroup Gamma1(13) of weight 2 over Rational Field
sage: ModularForms(Gamma1(13),2).dimension()
13
sage: [ModularForms(Gamma1(7),k).dimension() for k in [2,3,4,5]]
[5, 7, 9, 11]
sage: ModularForms(Gamma1(5),11).dimension()
12
We create a space with character::
sage: e = (DirichletGroup(13).0)^2
sage: e.order()
6
sage: M = ModularForms(e, 2); M
Modular Forms space of dimension 3, character [zeta6] and weight 2 over Cyclotomic Field of order 6 and degree 2
sage: f = M.T(2).charpoly('x'); f
x^3 + (-2*zeta6 - 2)*x^2 - 2*zeta6*x + 14*zeta6 - 7
sage: f.factor()
(x - zeta6 - 2) * (x - 2*zeta6 - 1) * (x + zeta6 + 1)
We can also create spaces corresponding to the groups `\Gamma_H(N)` intermediate
between `\Gamma_0(N)` and `\Gamma_1(N)`::
sage: G = GammaH(30, [11])
sage: M = ModularForms(G, 2); M
Modular Forms space of dimension 20 for Congruence Subgroup Gamma_H(30) with H generated by [11] of weight 2 over Rational Field
sage: M.T(7).charpoly().factor() # long time (7s on sage.math, 2011)
(x + 4) * x^2 * (x - 6)^4 * (x + 6)^4 * (x - 8)^7 * (x^2 + 4)
More examples of spaces with character::
sage: e = DirichletGroup(5, RationalField()).gen(); e
Dirichlet character modulo 5 of conductor 5 mapping 2 |--> -1
sage: m = ModularForms(e, 2); m
Modular Forms space of dimension 2, character [-1] and weight 2 over Rational Field
sage: m == loads(dumps(m))
True
sage: m.T(2).charpoly('x')
x^2 - 1
sage: m = ModularForms(e, 6); m.dimension()
4
sage: m.T(2).charpoly('x')
x^4 - 917*x^2 - 42284
This came up in a subtle bug (trac #5923)::
sage: ModularForms(gp(1), gap(12))
Modular Forms space of dimension 2 for Modular Group SL(2,Z) of weight 12 over Rational Field
This came up in another bug (related to trac #8630)::
sage: chi = DirichletGroup(109, CyclotomicField(3)).0
sage: ModularForms(chi, 2, base_ring = CyclotomicField(15))
Modular Forms space of dimension 10, character [zeta3 + 1] and weight 2 over Cyclotomic Field of order 15 and degree 8
We create some weight 1 spaces. The first example works fine, since we can prove purely by Riemann surface theory that there are no weight 1 cusp forms::
sage: M = ModularForms(Gamma1(11), 1); M
Modular Forms space of dimension 5 for Congruence Subgroup Gamma1(11) of weight 1 over Rational Field
sage: M.basis()
[
1 + 22*q^5 + O(q^6),
q + 4*q^5 + O(q^6),
q^2 - 4*q^5 + O(q^6),
q^3 - 5*q^5 + O(q^6),
q^4 - 3*q^5 + O(q^6)
]
sage: M.cuspidal_subspace().basis()
[
]
sage: M == M.eisenstein_subspace()
True
This example doesn't work so well, because we can't calculate the cusp
forms; but we can still work with the Eisenstein series.
sage: M = ModularForms(Gamma1(57), 1); M
Modular Forms space of dimension (unknown) for Congruence Subgroup Gamma1(57) of weight 1 over Rational Field
sage: M.basis()
Traceback (most recent call last):
...
NotImplementedError: Computation of dimensions of weight 1 cusp forms spaces not implemented in general
sage: M.cuspidal_subspace().basis()
Traceback (most recent call last):
...
NotImplementedError: Computation of dimensions of weight 1 cusp forms spaces not implemented in general
sage: E = M.eisenstein_subspace(); E
Eisenstein subspace of dimension 36 of Modular Forms space of dimension (unknown) for Congruence Subgroup Gamma1(57) of weight 1 over Rational Field
sage: (E.0 + E.2).q_expansion(40)
1 + q^2 + 1473/2*q^36 - 1101/2*q^37 + q^38 - 373/2*q^39 + O(q^40)
"""
if isinstance(group, dirichlet.DirichletCharacter):
if base_ring is None:
base_ring = group.minimize_base_ring().base_ring()
if base_ring is None:
base_ring = rings.QQ
if isinstance(group, dirichlet.DirichletCharacter) \
or arithgroup.is_CongruenceSubgroup(group):
level = group.level()
else:
level = group
key = canonical_parameters(group, level, weight, base_ring)
if use_cache and _cache.has_key(key):
M = _cache[key]()
if not (M is None):
M.set_precision(prec)
return M
(level, group, weight, base_ring) = key
M = None
if arithgroup.is_Gamma0(group):
M = ambient_g0.ModularFormsAmbient_g0_Q(group.level(), weight)
if base_ring != rings.QQ:
M = ambient_R.ModularFormsAmbient_R(M, base_ring)
elif arithgroup.is_Gamma1(group):
M = ambient_g1.ModularFormsAmbient_g1_Q(group.level(), weight)
if base_ring != rings.QQ:
M = ambient_R.ModularFormsAmbient_R(M, base_ring)
elif arithgroup.is_GammaH(group):
M = ambient_g1.ModularFormsAmbient_gH_Q(group, weight)
if base_ring != rings.QQ:
M = ambient_R.ModularFormsAmbient_R(M, base_ring)
elif isinstance(group, dirichlet.DirichletCharacter):
eps = group
if eps.base_ring().characteristic() != 0:
# TODO -- implement this
# Need to add a lift_to_char_0 function for characters,
# and need to still remember eps.
raise NotImplementedError, "currently the character must be over a ring of characteristic 0."
eps = eps.minimize_base_ring()
if eps.is_trivial():
return ModularForms(eps.modulus(), weight, base_ring,
use_cache = use_cache,
prec = prec)
M = ambient_eps.ModularFormsAmbient_eps(eps, weight)
if base_ring != eps.base_ring():
M = M.base_extend(base_ring) # ambient_R.ModularFormsAmbient_R(M, base_ring)
if M is None:
raise NotImplementedError, \
"computation of requested space of modular forms not defined or implemented"
M.set_precision(prec)
_cache[key] = weakref.ref(M)
return M
