def dimension_new_cusp_forms(self, k=2, eps=None, p=0, algorithm="CohenOesterle"):
r"""
Dimension of the new subspace (or `p`-new subspace) of cusp forms of
weight `k` and character `\varepsilon`.
INPUT:
- ``k`` - an integer (default: 2)
- ``eps`` - a Dirichlet character
- ``p`` - a prime (default: 0); just the `p`-new subspace if given
- ``algorithm`` - either "CohenOesterle" (the default) or "Quer". This
specifies the method to use in the case of nontrivial character:
either the Cohen--Oesterle formula as described in Stein's book, or
by Moebius inversion using the subgroups GammaH (a method due to
Jordi Quer).
EXAMPLES::
sage: G = DirichletGroup(9)
sage: eps = G.0^3
sage: eps.conductor()
3
sage: [Gamma1(9).dimension_new_cusp_forms(k, eps) for k in [2..10]]
[0, 0, 0, 2, 0, 2, 0, 2, 0]
sage: [Gamma1(9).dimension_cusp_forms(k, eps) for k in [2..10]]
[0, 0, 0, 2, 0, 4, 0, 6, 0]
sage: [Gamma1(9).dimension_new_cusp_forms(k, eps, 3) for k in [2..10]]
[0, 0, 0, 2, 0, 2, 0, 2, 0]
Double check using modular symbols (independent calculation)::
sage: [ModularSymbols(eps,k,sign=1).cuspidal_subspace().new_subspace().dimension() for k in [2..10]]
[0, 0, 0, 2, 0, 2, 0, 2, 0]
sage: [ModularSymbols(eps,k,sign=1).cuspidal_subspace().new_subspace(3).dimension() for k in [2..10]]
[0, 0, 0, 2, 0, 2, 0, 2, 0]
Another example at level 33::
sage: G = DirichletGroup(33)
sage: eps = G.1
sage: eps.conductor()
11
sage: [Gamma1(33).dimension_new_cusp_forms(k, G.1) for k in [2..4]]
[0, 4, 0]
sage: [Gamma1(33).dimension_new_cusp_forms(k, G.1, algorithm="Quer") for k in [2..4]]
[0, 4, 0]
sage: [Gamma1(33).dimension_new_cusp_forms(k, G.1^2) for k in [2..4]]
[2, 0, 6]
sage: [Gamma1(33).dimension_new_cusp_forms(k, G.1^2, p=3) for k in [2..4]]
[2, 0, 6]
"""
if eps == None:
return GammaH_class.dimension_new_cusp_forms(self, k, p)
N = self.level()
eps = DirichletGroup(N)(eps)
from all import Gamma0
if eps.is_trivial():
return Gamma0(N).dimension_new_cusp_forms(k, p)
from congroup_gammaH import mumu
if p == 0 or N%p != 0 or eps.conductor().valuation(p) == N.valuation(p):
D = [eps.conductor()*d for d in divisors(N//eps.conductor())]
return sum([Gamma1_constructor(M).dimension_cusp_forms(k, eps.restrict(M), algorithm)*mumu(N//M) for M in D])
eps_p = eps.restrict(N//p)
old = Gamma1_constructor(N//p).dimension_cusp_forms(k, eps_p, algorithm)
return self.dimension_cusp_forms(k, eps, algorithm) - 2*old
def dimension_new_cusp_forms(self,
k=2,
eps=None,
p=0,
algorithm="CohenOesterle"):
r"""
Dimension of the new subspace (or `p`-new subspace) of cusp forms of
weight `k` and character `\varepsilon`.
INPUT:
- ``k`` - an integer (default: 2)
- ``eps`` - a Dirichlet character
- ``p`` - a prime (default: 0); just the `p`-new subspace if given
- ``algorithm`` - either "CohenOesterle" (the default) or "Quer". This
specifies the method to use in the case of nontrivial character:
either the Cohen--Oesterle formula as described in Stein's book, or
by Moebius inversion using the subgroups GammaH (a method due to
Jordi Quer).
EXAMPLES::
sage: G = DirichletGroup(9)
sage: eps = G.0^3
sage: eps.conductor()
3
sage: [Gamma1(9).dimension_new_cusp_forms(k, eps) for k in [2..10]]
[0, 0, 0, 2, 0, 2, 0, 2, 0]
sage: [Gamma1(9).dimension_cusp_forms(k, eps) for k in [2..10]]
[0, 0, 0, 2, 0, 4, 0, 6, 0]
sage: [Gamma1(9).dimension_new_cusp_forms(k, eps, 3) for k in [2..10]]
[0, 0, 0, 2, 0, 2, 0, 2, 0]
Double check using modular symbols (independent calculation)::
sage: [ModularSymbols(eps,k,sign=1).cuspidal_subspace().new_subspace().dimension() for k in [2..10]]
[0, 0, 0, 2, 0, 2, 0, 2, 0]
sage: [ModularSymbols(eps,k,sign=1).cuspidal_subspace().new_subspace(3).dimension() for k in [2..10]]
[0, 0, 0, 2, 0, 2, 0, 2, 0]
Another example at level 33::
sage: G = DirichletGroup(33)
sage: eps = G.1
sage: eps.conductor()
11
sage: [Gamma1(33).dimension_new_cusp_forms(k, G.1) for k in [2..4]]
[0, 4, 0]
sage: [Gamma1(33).dimension_new_cusp_forms(k, G.1, algorithm="Quer") for k in [2..4]]
[0, 4, 0]
sage: [Gamma1(33).dimension_new_cusp_forms(k, G.1^2) for k in [2..4]]
[2, 0, 6]
sage: [Gamma1(33).dimension_new_cusp_forms(k, G.1^2, p=3) for k in [2..4]]
[2, 0, 6]
"""
if eps == None:
return GammaH_class.dimension_new_cusp_forms(self, k, p)
N = self.level()
eps = DirichletGroup(N)(eps)
from all import Gamma0
if eps.is_trivial():
return Gamma0(N).dimension_new_cusp_forms(k, p)
from congroup_gammaH import mumu
if p == 0 or N % p != 0 or eps.conductor().valuation(p) == N.valuation(
p):
D = [eps.conductor() * d for d in divisors(N // eps.conductor())]
return sum([
Gamma1_constructor(M).dimension_cusp_forms(
k, eps.restrict(M), algorithm) * mumu(N // M) for M in D
])
eps_p = eps.restrict(N // p)
old = Gamma1_constructor(N // p).dimension_cusp_forms(
k, eps_p, algorithm)
return self.dimension_cusp_forms(k, eps, algorithm) - 2 * old
