group, weight, character, sign, base_ring) |
character,
cuspidal_submodule,
cuspidal_subspace,
dual_star_involution_matrix,
eisenstein_subspace,
factorization,
group,
hecke_bound,
hecke_module_of_level,
is_cuspidal,
is_simple,
minus_subspace,
multiplicity,
new_subspace,
ngens,
plus_subspace,
q_expansion,
q_expansion_basis,
sign,
sign_subspace,
simple_factors,
star_decomposition,
star_eigenvalues,
star_involution,
sturm_bound
Further documentation:
) |
Synonym for cuspidal_submodule.
) |
Return the matrix of the dual star involution, which is induced by complex conjugation on the linear dual of modular symbols.
) |
Synonym for eisenstein_submodule.
) |
Returns a list of pairs
where
is simple spaces of
modular symbols and self is isomorphic to the direct sum of
the
as a module over the anemic Hecke algebra
adjoin the star involution.
ASSUMPTION: self is a module over the anemic Hecke algebra.
) |
Returns the group of this modular symbols space.
INPUT: ModularSymbols self -- an arbitrary space of modular symbols OUTPUT: CongruenceSubgroup -- the congruence subgroup that this is a space of modular symbols for.
ALGORITHM: The group is recorded when this space is created.
sage: m = ModularSymbols(20) sage: m.group() Gamma0(20)
) |
Returns the Sturm bound for this space of modular symbols.
Returns a positive integer
such that the Hecke operators
acting on cusp forms generate the
Hecke algebra as a
-module when the character is trivial
or quadratic. Otherwise,
generate the Hecke
algebra at least as a
-module, where
is
the ring generated by the values of the Dirichlet character
. Alternatively, this is a bound such that if two cusp
forms associated to this space of modular symbols are
congruent modulo
, then they are congruent
modulo
.
REFERENCES: See the Agashe-Stein appendix to Lario and Schoof's Some computations with Hecke rings and deformation rings, Experimental Math., 11 (2002), no. 2, 303-311. This result originated in the paper Sturm, On the congruence of modular forms, Springer LNM 1240, 275-280, 1987.
REMARK:
Kevin Buzzard pointed out to me (William Stein) in Fall 2002
that the above bound is fine for
with character,
as one sees by taking a power of
. More precisely, if
for first
coefficients, then
for first
coefficents. Since the weight of
is
, it follows that if
, where
is the Sturm bound for
at weight
, then
has valuation large enough to be forced to be 0
at
by Sturm bound (which is valid if we choose
correctly). Thus
. Conclusion: For
with fixed character, the Sturm bound is
exactly the same as for
.
A key point is that we are finding
generators for
the Hecke algebra here, not
-generators. So if one wants
generators for the Hecke algebra over
, this bound must
be suitably modified.
level) |
See the documentation for self.modular_symbols_of_level(level)
[compute_dual=True]) |
Return the subspace of self on which the star involution acts as -1.
INPUT: compute_dual -- bool (default: True) also compute dual subspace. This are useful for many algorithms. OUTPUT: subspace of modular symbols
S) |
Return the multiplicity of the simple modular symbols space S in self. S must be a simple anemic Hecke module.
ASSUMPTION: self is an anemic Hecke module with the same weight and group as S, and S is simple.
[p=None]) |
Synonym for new_submodule.
) |
The number of generators of self.
INPUT: ModularSymbols self -- arbitrary space of modular symbols. OUTPUT: int -- the number of generators, which is the same as the dimension of self. ALGORITHM: Call the dimension function.
sage: m = ModularSymbols(33) sage: m.ngens() 9 sage: m.rank() 9 sage: ModularSymbols(100, weight=2, sign=1).ngens() 18
[compute_dual=True]) |
Return the subspace of self on which the star involution acts as +1.
INPUT: compute_dual -- bool (default: True) also compute dual subspace. This are useful for many algorithms. OUTPUT: subspace of modular symbols
prec) |
Returns the q-expansion to precision prec of a newform associated to self, where self must be new, cuspidal, and simple.
prec) |
Returns a basis of q-expansions (as power series) to precision prec of the space of modular forms associated to self. The q-expansions are defined over the same base ring as prec.
) |
Returns the sign of self.
For efficiency reasons, it is often useful to compute in the (largest) quotient of modular symbols where the * involution acts as +1, or where it acts as -1.
INPUT: ModularSymbols self -- arbitrary space of modular symbols. OUTPUT: int -- the sign of self, either -1, 0, or 1. -1 -- largest quotient where * acts as -1, +1 -- largest quotient where * acts as +1, 0 -- full space of modular symbols (no quotient).
ALGORITHM: Call the dimension function.
sage: m = ModularSymbols(33) sage: m.rank() 9 sage: m.sign() 0 sage: m = ModularSymbols(33, sign=0) sage: m.sign() 0 sage: m.rank() 9 sage: m = ModularSymbols(33, sign=-1) sage: m.sign() -1 sage: m.rank() 3
sign, [compute_dual=True]) |
Return the subspace of self that is fixed under the star involution.
INPUT: sign -- int (either -1, 0 or +1) compute_dual -- bool (default: True) also compute dual subspace. This are useful for many algorithms. OUTPUT: subspace of modular symbols
) |
Returns a list modular symbols spaces
where
is simple
spaces of modular symbols (for the anemic Hecke algebra) and
self is isomorphic to the direct sum of the
with some
multiplicities, as a module over the anemic Hecke
algebra. For the multiplicities use factorization() instead.
ASSUMPTION: self is a module over the anemic Hecke algebra.
) |
Returns the eigenvalues of the star involution acting on self.
sage: M = ModularSymbols(11) sage: D = M.decomposition() sage: M.star_eigenvalues() [1, -1] sage: D[0].star_eigenvalues() [1] sage: D[1].star_eigenvalues() [1, -1] sage: D[1].plus_subspace().star_eigenvalues() [1] sage: D[1].minus_subspace().star_eigenvalues() [-1]
) |
Return the star involution on self, which is induced by complex conjugation on modular symbols.
) |
Returns the Sturm bound for this space of modular symbols.
Returns a positive integer
such that the Hecke operators
acting on cusp forms generate the
Hecke algebra as a
-module when the character is trivial
or quadratic. Otherwise,
generate the Hecke
algebra at least as a
-module, where
is
the ring generated by the values of the Dirichlet character
. Alternatively, this is a bound such that if two cusp
forms associated to this space of modular symbols are
congruent modulo
, then they are congruent
modulo
.
REFERENCES: See the Agashe-Stein appendix to Lario and Schoof, Some computations with Hecke rings and deformation rings, Experimental Math., 11 (2002), no. 2, 303-311. This result originated in the paper Sturm, On the congruence of modular forms, Springer LNM 1240, 275-280, 1987.
REMARK:
Kevin Buzzard pointed out to me (William Stein) in Fall 2002
that the above bound is fine for
with character,
as one sees by taking a power of
. More precisely, if
for first
coefficients, then
for first
coefficents. Since the weight of
is
, it follows that if
, where
is the Sturm bound for
at weight
, then
has valuation large enough to be forced to be 0
at
by Sturm bound (which is valid if we choose
correctly). Thus
. Conclusion: For
with fixed character, the Sturm bound is
exactly the same as for
.
A key point is that we are finding
generators for
the Hecke algebra here, not
-generators. So if one wants
generators for the Hecke algebra over
, this bound must
be suitably modified.
Instances of class ModularSymbolsSpace also have the following special methods:
__cmp__
Further documentation:
other) |
Compare self and other.