def modular_symbols(self):
"""
Return the space of modular symbols used for computing this
set of modular eigenforms.
EXAMPLES::
sage: n = numerical_eigenforms(61) ; n.modular_symbols()
Modular Symbols space of dimension 5 for Gamma_0(61) of weight 2 with sign 1 over Rational Field
"""
M = ModularSymbols(self._group, self._weight, sign=1)
if M.base_ring() != QQ:
raise ValueError("modular forms space must be defined over QQ")
return M
def modular_symbol_space(E, sign, base_ring, bound=None):
r"""
Creates the space of modular symbols of a given sign over a give base_ring,
attached to the isogeny class of elliptic curves.
INPUT:
- ``E`` - an elliptic curve over `\QQ`
- ``sign`` - integer, -1, 0, or 1
- ``base_ring`` - ring
- ``bound`` - (default: None) maximum number of Hecke operators to
use to cut out modular symbols factor. If None, use
enough to provably get the correct answer.
OUTPUT: a space of modular symbols
EXAMPLES::
sage: import sage.schemes.elliptic_curves.ell_modular_symbols
sage: E=EllipticCurve('11a1')
sage: M=sage.schemes.elliptic_curves.ell_modular_symbols.modular_symbol_space(E,-1,GF(37))
sage: M
Modular Symbols space of dimension 1 for Gamma_0(11) of weight 2 with sign -1 over Finite Field of size 37
"""
_sign = int(sign)
if _sign != sign:
raise TypeError('sign must be an integer')
if not (_sign in [-1, 0, 1]):
raise TypeError('sign must -1, 0, or 1')
N = E.conductor()
M = ModularSymbols(N, sign=sign, base_ring=base_ring)
if bound is None:
bound = M.hecke_bound() + 10
V = M
p = 2
while p <= bound and V.dimension() > 1:
t = V.T(p)
ap = E.ap(p)
V = (t - ap).kernel()
p = next_prime(p)
return V
def modular_symbols(self):
"""
Return the space of modular symbols used for computing this
set of modular eigenforms.
EXAMPLES::
sage: n = numerical_eigenforms(61) ; n.modular_symbols()
Modular Symbols space of dimension 5 for Gamma_0(61) of weight 2 with sign 1 over Rational Field
"""
try:
return self.__modular_symbols
except AttributeError:
M = ModularSymbols(self._group,
self._weight, sign=1)
if M.base_ring() != QQ:
raise ValueError("modular forms space must be defined over QQ")
self.__modular_symbols = M
return M
def modular_symbol_space(E, sign, base_ring, bound=None):
r"""
Creates the space of modular symbols of a given sign over a give base_ring,
attached to the isogeny class of elliptic curves.
INPUT:
- ``E`` - an elliptic curve over `\QQ`
- ``sign`` - integer, -1, 0, or 1
- ``base_ring`` - ring
- ``bound`` - (default: None) maximum number of Hecke operators to
use to cut out modular symbols factor. If None, use
enough to provably get the correct answer.
OUTPUT: a space of modular symbols
EXAMPLES::
sage: import sage.schemes.elliptic_curves.ell_modular_symbols
sage: E=EllipticCurve('11a1')
sage: M=sage.schemes.elliptic_curves.ell_modular_symbols.modular_symbol_space(E,-1,GF(37))
sage: M
Modular Symbols space of dimension 1 for Gamma_0(11) of weight 2 with sign -1 over Finite Field of size 37
"""
_sign = int(sign)
if _sign != sign:
raise TypeError, 'sign must be an integer'
if not (_sign in [-1,0,1]):
raise TypeError, 'sign must -1, 0, or 1'
N = E.conductor()
M = ModularSymbols(N, sign=sign, base_ring=base_ring)
if bound is None:
bound = M.hecke_bound() + 10
V = M
p = 2
while p <= bound and V.dimension() > 1:
t = V.T(p)
ap = E.ap(p)
V = (t - ap).kernel()
p = next_prime(p)
return V
