def create_key(self, k, p=None, prec_cap=None, base=None, base_coeffs=None, \
character=None, adjuster=None, act_on_left=False, \
dettwist=None, variable_name = 'w'):
if base is None:
if base_coeffs is None:
if p is None:
raise ValueError("Must specify a prime, a base ring, or coefficients.")
if prec_cap is None:
raise ValueError("Must specify a precision cap or a base ring.")
prec_cap = _prec_cap_parser(prec_cap)
base_coeffs = Zp(p, prec = prec_cap[0])
elif not isinstance(base_coeffs, ring.Ring):
raise TypeError("base_coeffs must be a ring if base is None")
elif prec_cap is None:
raise ValueError("Must specify a precision cap or a base ring.")
else:
prec_cap = _prec_cap_parser(prec_cap)
if base_coeffs.precision_cap() < prec_cap[0]:
raise ValueError("Precision cap on coefficients of base ring must be at least the p-adic precision cap of this space.")
base = PowerSeriesRing(base_coeffs, name=variable_name, default_prec=prec_cap[1])
elif prec_cap is None:
prec_cap = [ZZ(base.base_ring().precision_cap()), ZZ(base.default_prec())]
else:
if base.base_ring().precision_cap() < prec_cap[0]:
raise ValueError("Precision cap on coefficients of base ring must be at least the p-adic precision cap of this space.")
if base.default_prec() < prec_cap[1]:
raise ValueError("Default precision on the variable of base ring must be at least the w-adic precision cap of this space.")
base_coeffs = None
p = base.base_ring().prime()
k_shift = 0
#if character is not None:
# #Should we ensure character is primitive?
# cond_val = character.conductor().valuation(p)
# if cond_val > 1:
# raise ValueError("Level must not be divisible by p^2.")
# elif cond_val == 1:
# pass
# else:
# pass
k = ZZ((k + k_shift) % (p-1))
#if prec_cap is None:
# prec_cap = [base.base_ring().precision_cap, base.default_prec()]
#else:
# prec_cap = list(prec_cap)
#prec_cap = [base.base_ring().precision_cap(), base.default_prec()]
if adjuster is None:
adjuster = _default_adjuster()
if dettwist is not None:
dettwist = ZZ(dettwist)
if dettwist == 0:
dettwist = None
return (k, p, tuple(prec_cap), base, character, adjuster, act_on_left, dettwist)
def eis_H(a, b, N, k, Q=None, t=1, prec=10):
if Q == None:
Q = PowerSeriesRing(CyclotomicField(N), 'q')
R = Q.base_ring()
zetaN = R.zeta(N)
q = Q.gen()
a = ZZ(a % N)
b = ZZ(b % N)
s = 0
if k == 1:
if a == 0 and not b == 0:
s = -QQ(1) / QQ(2) * (1 + zetaN**b) / (1 - zetaN**b)
elif b == 0 and not a == 0:
s = -QQ(1) / QQ(2) * (1 + zetaN**a) / (1 - zetaN**a)
elif a != 0 and b != 0:
s = -QQ(1) / QQ(2) * ((1 + zetaN**a) / (1 - zetaN**a) +
(1 + zetaN**b) / (1 - zetaN**b))
elif k > 1:
s = hurwitz_hat(-b, N, 1 - k, zetaN)
for m in srange(1, prec / t):
for n in srange(1, prec / t / m + 1):
s += (zetaN**(-a * m - b * n) +
(-1)**k * zetaN**(a * m + b * n)) * n**(k - 1) * q**(m * n)
return s + O(q**floor(prec))
def eis_G(a, b, N, k, Q=None, t=1, prec=20):
if Q == None:
Q = PowerSeriesRing(QQ, 'q')
R = Q.base_ring()
q = Q.gen()
a = ZZ(a % N)
b = ZZ(b % N)
s = 0
if k == 1:
if a == 0 and not b == 0:
s = QQ(1) / QQ(2) - QQ(b % N) / QQ(N)
elif b == 0 and not a == 0:
s = QQ(1) / QQ(2) - QQ(a % N) / QQ(N)
elif k > 1:
if b == 0:
s = -N**(k - 1) * ber_pol(QQ(a % N) / QQ(N), k) / QQ(k)
#If a == 0 or b ==0 the loop has to start at 1
starta, startb = 0, 0
if a == 0:
starta = 1
if b == 0:
startb = 1
for v in srange(starta, (prec / t + a) / N):
for w in srange(startb, (prec / t / abs((-a + v * N)) + b) / N + 1):
s += q**(t * (a + v * N) * (b + w * N)) * (a + v * N)**(k - 1)
if (-a + v * N) > 0 and (-b + w * N) > 0:
s += (-1)**k * q**(t * (-a + v * N) *
(-b + w * N)) * (-a + v * N)**(k - 1)
return s + O(q**floor(prec))
def eis_E(cv, dv, N, k, Q=None, param_level=1, prec=10):
r"""
Computes the coefficient of the Eisenstein series for $\Gamma(N)$.
Not intended to be called by user.
INPUT:
- cv - int, the first coordinate of the vector determining the \Gamma(N)
Eisenstein series
- dv - int, the second coordinate of the vector determining the \Gamma(N)
Eisenstein series
- N - int, the level of the Eisenstein series to be computed
- k - int, the weight of the Eisenstein seriess to be computed
- Q - power series ring, the ring containing the q-expansion to be computed
- param_level - int, the parameter of the returned series will be
q_{param_level}
- prec - int, the precision. The series in q_{param_level} will be truncated
after prec coefficients
OUTPUT:
- an element of the ring Q, which is the Fourier expansion of the Eisenstein
series
"""
if Q == None:
Q = PowerSeriesRing(CyclotomicField(N), 'q')
R = Q.base_ring()
zetaN = R.zeta(N)
q = Q.gen()
cv = cv % N
dv = dv % N
#if dv == 0 and cv == 0 and k == 2:
# raise ValueError("E_2 is not a modular form")
if k == 1:
if cv == 0 and dv == 0:
raise ValueError("that shouldn't have happened...")
elif cv == 0 and dv != 0:
s = QQ(1) / QQ(2) * (1 + zetaN**dv) / (1 - zetaN**dv)
elif cv != 0:
s = QQ(1) / QQ(2) - QQ(cv) / QQ(N) + floor(QQ(cv) / QQ(N))
elif k > 1:
if cv == 0:
s = hurwitz_hat(QQ(dv), QQ(N), 1 - k, zetaN)
else:
s = 0
for n1 in xrange(1, prec): # this is n/m in DS
for n2 in xrange(1, prec / n1 + 1): # this is m in DS
if Mod(n1, N) == Mod(cv, N):
s += n2**(k - 1) * zetaN**(dv * n2) * q**(n1 * n2)
if Mod(n1, N) == Mod(-cv, N):
s += (-1)**k * n2**(k - 1) * zetaN**(-dv * n2) * q**(n1 * n2)
return s + O(q**floor(prec))
def eis_F(cv, dv, N, k, Q=None, prec=10, t=1):
"""
Computes the coefficient of the Eisenstein series for $\Gamma(N)$.
Not indented to be called by user.
INPUT:
- cv - int, the first coordinate of the vector determining the \Gamma(N)
Eisenstein series
- dv - int, the second coordinate of the vector determining the \Gamma(N)
Eisenstein series
- N - int, the level of the Eisenstein series to be computed
- k - int, the weight of the Eisenstein seriess to be computed
- Q - power series ring, the ring containing the q-expansion to be computed
- param_level - int, the parameter of the returned series will be
q_{param_level}
- prec - int, the precision. The series in q_{param_level} will be truncated
after prec coefficients
OUTPUT:
- an element of the ring Q, which is the Fourier expansion of the Eisenstein
series
"""
if Q == None:
Q = PowerSeriesRing(CyclotomicField(N), 'q{}'.format(N))
R = Q.base_ring()
zetaN = R.zeta(N)
q = Q.gen()
s = 0
if k == 1:
if cv % N == 0 and dv % N != 0:
s = QQ(1) / QQ(2) * (1 + zetaN**dv) / (1 - zetaN**dv)
elif cv % N != 0:
s = QQ(1) / QQ(2) - QQ(cv) / QQ(N) + floor(QQ(cv) / QQ(N))
elif k > 1:
s = -ber_pol(QQ(cv) / QQ(N) - floor(QQ(cv) / QQ(N)), k) / QQ(k)
for n1 in xrange(1, ceil(prec / QQ(t))): # this is n/m in DS
for n2 in xrange(1,
ceil(prec / QQ(t) / QQ(n1)) + 1): # this is m in DS
if Mod(n1, N) == Mod(cv, N):
s += N**(1 - k) * n1**(k - 1) * zetaN**(dv * n2) * q**(t * n1 *
n2)
if Mod(n1, N) == Mod(-cv, N):
s += (-1)**k * N**(1 - k) * n1**(k - 1) * zetaN**(
-dv * n2) * q**(t * n1 * n2)
return s + O(q**floor(prec))
def __init__(self, k, p=None, prec_cap=[20, 10], base=None, base_coeffs=None, \
character=None, adjuster=None, act_on_left=False, \
dettwist=None, action_class = WeightKAction_OMS_fam, \
variable_name = 'w'):
#self._prec_cap = prec_cap
if base is None:
if base_coeffs is None:
base_coeffs = Zp(p, prec = prec_cap[0])
elif not isinstance(base_coeffs, ring.Ring):
raise TypeError("base_coeffs must be a ring if base is None")
base = PowerSeriesRing(base_coeffs, name=variable_name)
#elif not isinstance(base, ring.Ring):
# raise TypeError("base must be a ring")
self._p = base.base_ring().prime()
#self._prec_cap = [base.base_ring().precision_cap(), base.default_prec()]
self._prec_cap = tuple(prec_cap)
k = k % (self._p - 1)
#self._cp = (self._p-2) / (self._p-1)
CoefficientModule_generic.__init__(self, k, base=base, \
character=character, adjuster=adjuster, act_on_left=act_on_left, \
dettwist=dettwist, action_class=action_class, \
element_class=CoeffMod_OMS_Families_element, padic=True)