def multiplicative_subgroups(self):
r"""
Return generators for each subgroup of
`(\ZZ/N\ZZ)^*`.
EXAMPLES::
sage: Integers(5).multiplicative_subgroups()
([2], [4], [])
sage: Integers(15).multiplicative_subgroups()
([11, 7], [4, 11], [8], [11], [14], [7], [4], [])
sage: Integers(2).multiplicative_subgroups()
([],)
sage: len(Integers(341).multiplicative_subgroups())
80
"""
from sage.groups.abelian_gps.abelian_group import AbelianGroup
from sage.misc.misc import mul
U = self.unit_gens()
G = AbelianGroup([x.multiplicative_order() for x in U])
rawsubs = G.subgroups()
mysubs = []
for G in rawsubs:
mysubs.append([])
for s in G.gens():
mysubs[-1].append(
mul([U[i]**s.list()[i] for i in xrange(len(U))]))
return tuple(mysubs) # make it immutable, so that we can cache
def multiplicative_subgroups(self):
r"""
Return generators for each subgroup of
`(\ZZ/N\ZZ)^*`.
EXAMPLES::
sage: Integers(5).multiplicative_subgroups()
([2], [4], [])
sage: Integers(15).multiplicative_subgroups()
([11, 7], [4, 11], [8], [11], [14], [7], [4], [])
sage: Integers(2).multiplicative_subgroups()
([],)
sage: len(Integers(341).multiplicative_subgroups())
80
"""
from sage.groups.abelian_gps.abelian_group import AbelianGroup
from sage.misc.misc import mul
U = self.unit_gens()
G = AbelianGroup([x.multiplicative_order() for x in U])
rawsubs = G.subgroups()
mysubs = []
for G in rawsubs:
mysubs.append([])
for s in G.gens():
mysubs[-1].append(mul([U[i] ** s.list()[i] for i in xrange(len(U))]))
return tuple(mysubs) # make it immutable, so that we can cache
def CohenOesterle(eps, k):
r"""
Compute the Cohen-Oesterle function associate to eps, `k`.
This is a summand in the formula for the dimension of the space of
cusp forms of weight `2` with character
`\varepsilon`.
INPUT:
- ``eps`` - Dirichlet character
- ``k`` - integer
OUTPUT: element of the base ring of eps.
EXAMPLES::
sage: G.<eps> = DirichletGroup(7)
sage: sage.modular.dims.CohenOesterle(eps, 2)
-2/3
sage: sage.modular.dims.CohenOesterle(eps, 4)
-1
"""
N = eps.modulus()
facN = factor(N)
f = eps.conductor()
gamma_k = 0
if k % 4 == 2:
gamma_k = frac(-1, 4)
elif k % 4 == 0:
gamma_k = frac(1, 4)
mu_k = 0
if k % 3 == 2:
mu_k = frac(-1, 3)
elif k % 3 == 0:
mu_k = frac(1, 3)
def _lambda(r, s, p):
"""
Used internally by the CohenOesterle function.
INPUT:
- ``r, s, p`` - integers
OUTPUT: Integer
EXAMPLES: (indirect doctest)
::
sage: K = CyclotomicField(3)
sage: eps = DirichletGroup(7*43,K).0^2
sage: sage.modular.dims.CohenOesterle(eps,2)
-4/3
"""
if 2 * s <= r:
if r % 2 == 0:
return p**(r // 2) + p**((r // 2) - 1)
return 2 * p**((r - 1) // 2)
return 2 * (p**(r - s))
#end def of lambda
K = eps.base_ring()
return K(frac(-1,2) * mul([_lambda(r,valuation(f,p),p) for p, r in facN]) + \
gamma_k * mul([CO_delta(r,p,N,eps) for p, r in facN]) + \
mu_k * mul([CO_nu(r,p,N,eps) for p, r in facN]))
def CohenOesterle(eps, k):
r"""
Compute the Cohen-Oesterle function associate to eps, `k`.
This is a summand in the formula for the dimension of the space of
cusp forms of weight `2` with character
`\varepsilon`.
INPUT:
- ``eps`` - Dirichlet character
- ``k`` - integer
OUTPUT: element of the base ring of eps.
EXAMPLES::
sage: G.<eps> = DirichletGroup(7)
sage: sage.modular.dims.CohenOesterle(eps, 2)
-2/3
sage: sage.modular.dims.CohenOesterle(eps, 4)
-1
"""
N = eps.modulus()
facN = factor(N)
f = eps.conductor()
gamma_k = 0
if k%4==2:
gamma_k = frac(-1,4)
elif k%4==0:
gamma_k = frac(1,4)
mu_k = 0
if k%3==2:
mu_k = frac(-1,3)
elif k%3==0:
mu_k = frac(1,3)
def _lambda(r,s,p):
"""
Used internally by the CohenOesterle function.
INPUT:
- ``r, s, p`` - integers
OUTPUT: Integer
EXAMPLES: (indirect doctest)
::
sage: K = CyclotomicField(3)
sage: eps = DirichletGroup(7*43,K).0^2
sage: sage.modular.dims.CohenOesterle(eps,2)
-4/3
"""
if 2*s<=r:
if r%2==0:
return p**(r//2) + p**((r//2)-1)
return 2*p**((r-1)//2)
return 2*(p**(r-s))
#end def of lambda
K = eps.base_ring()
return K(frac(-1,2) * mul([_lambda(r,valuation(f,p),p) for p, r in facN]) + \
gamma_k * mul([CO_delta(r,p,N,eps) for p, r in facN]) + \
mu_k * mul([CO_nu(r,p,N,eps) for p, r in facN]))
