def eisen(p):
"""
Return the Eisenstein number `n` which is the numerator of `(p-1)/12`.
INPUT:
- ``p`` -- a prime
OUTPUT: Integer
EXAMPLES::
sage: [(p, sage.modular.dims.eisen(p)) for p in prime_range(24)]
[(2, 1), (3, 1), (5, 1), (7, 1), (11, 5), (13, 1), (17, 4),
(19, 3), (23, 11)]
"""
if not is_prime(p):
raise ValueError("p must be prime")
return frac(p - 1, 12).numerator()
def eisen(p):
"""
Return the Eisenstein number `n` which is the numerator of
`(p-1)/12`.
INPUT:
- ``p`` - a prime
OUTPUT: Integer
EXAMPLES::
sage: [(p,sage.modular.dims.eisen(p)) for p in prime_range(24)]
[(2, 1), (3, 1), (5, 1), (7, 1), (11, 5), (13, 1), (17, 4), (19, 3), (23, 11)]
"""
if not is_prime(p):
raise ValueError, "p must be prime"
return frac(p-1,12).numerator()
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]))
