Ambient Spaces of Modular Forms¶
EXAMPLES:
We compute a basis for the ambient space \(M_2(\Gamma_1(25),\chi)\), where \(\chi\) is quadratic.
sage: chi = DirichletGroup(25,QQ).0; chi
Dirichlet character modulo 25 of conductor 5 mapping 2 |--> -1
sage: n = ModularForms(chi,2); n
Modular Forms space of dimension 6, character [-1] and weight 2 over Rational Field
sage: type(n)
<class 'sage.modular.modform.ambient_eps.ModularFormsAmbient_eps_with_category'>
Compute a basis:
sage: n.basis()
[
1 + O(q^6),
q + O(q^6),
q^2 + O(q^6),
q^3 + O(q^6),
q^4 + O(q^6),
q^5 + O(q^6)
]
Compute the same basis but to higher precision:
sage: n.set_precision(20)
sage: n.basis()
[
1 + 10*q^10 + 20*q^15 + O(q^20),
q + 5*q^6 + q^9 + 12*q^11 - 3*q^14 + 17*q^16 + 8*q^19 + O(q^20),
q^2 + 4*q^7 - q^8 + 8*q^12 + 2*q^13 + 10*q^17 - 5*q^18 + O(q^20),
q^3 + q^7 + 3*q^8 - q^12 + 5*q^13 + 3*q^17 + 6*q^18 + O(q^20),
q^4 - q^6 + 2*q^9 + 3*q^14 - 2*q^16 + 4*q^19 + O(q^20),
q^5 + q^10 + 2*q^15 + O(q^20)
]
- class sage.modular.modform.ambient.ModularFormsAmbient(group, weight, base_ring, character=None, eis_only=False)¶
Bases:
sage.modular.modform.space.ModularFormsSpace,sage.modular.hecke.ambient_module.AmbientHeckeModuleAn ambient space of modular forms.
- ambient_space()¶
Return the ambient space that contains this ambient space. This is, of course, just this space again.
EXAMPLES:
sage: m = ModularForms(Gamma0(3),30) sage: m.ambient_space() is m True
- change_ring(base_ring)¶
Change the base ring of this space of modular forms.
INPUT:
R- ring
EXAMPLES:
sage: M = ModularForms(Gamma0(37),2) sage: M.basis() [ q + q^3 - 2*q^4 + O(q^6), q^2 + 2*q^3 - 2*q^4 + q^5 + O(q^6), 1 + 2/3*q + 2*q^2 + 8/3*q^3 + 14/3*q^4 + 4*q^5 + O(q^6) ]
The basis after changing the base ring is the reduction modulo \(3\) of an integral basis.
sage: M3 = M.change_ring(GF(3)) sage: M3.basis() [ q + q^3 + q^4 + O(q^6), q^2 + 2*q^3 + q^4 + q^5 + O(q^6), 1 + q^3 + q^4 + 2*q^5 + O(q^6) ]
- cuspidal_submodule()¶
Return the cuspidal submodule of this ambient module.
EXAMPLES:
sage: ModularForms(Gamma1(13)).cuspidal_submodule() Cuspidal subspace of dimension 2 of Modular Forms space of dimension 13 for Congruence Subgroup Gamma1(13) of weight 2 over Rational Field
- dimension()¶
Return the dimension of this ambient space of modular forms, computed using a dimension formula (so it should be reasonably fast).
EXAMPLES:
sage: m = ModularForms(Gamma1(20),20) sage: m.dimension() 238
- eisenstein_params()¶
Return parameters that define all Eisenstein series in self.
OUTPUT: an immutable Sequence
EXAMPLES:
sage: m = ModularForms(Gamma0(22), 2) sage: v = m.eisenstein_params(); v [(Dirichlet character modulo 22 of conductor 1 mapping 13 |--> 1, Dirichlet character modulo 22 of conductor 1 mapping 13 |--> 1, 2), (Dirichlet character modulo 22 of conductor 1 mapping 13 |--> 1, Dirichlet character modulo 22 of conductor 1 mapping 13 |--> 1, 11), (Dirichlet character modulo 22 of conductor 1 mapping 13 |--> 1, Dirichlet character modulo 22 of conductor 1 mapping 13 |--> 1, 22)] sage: type(v) <class 'sage.structure.sequence.Sequence_generic'>
- eisenstein_series()¶
Return all Eisenstein series associated to this space.
sage: ModularForms(27,2).eisenstein_series() [ q^3 + O(q^6), q - 3*q^2 + 7*q^4 - 6*q^5 + O(q^6), 1/12 + q + 3*q^2 + q^3 + 7*q^4 + 6*q^5 + O(q^6), 1/3 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + O(q^6), 13/12 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + O(q^6) ]
sage: ModularForms(Gamma1(5),3).eisenstein_series() [ -1/5*zeta4 - 2/5 + q + (4*zeta4 + 1)*q^2 + (-9*zeta4 + 1)*q^3 + (4*zeta4 - 15)*q^4 + q^5 + O(q^6), q + (zeta4 + 4)*q^2 + (-zeta4 + 9)*q^3 + (4*zeta4 + 15)*q^4 + 25*q^5 + O(q^6), 1/5*zeta4 - 2/5 + q + (-4*zeta4 + 1)*q^2 + (9*zeta4 + 1)*q^3 + (-4*zeta4 - 15)*q^4 + q^5 + O(q^6), q + (-zeta4 + 4)*q^2 + (zeta4 + 9)*q^3 + (-4*zeta4 + 15)*q^4 + 25*q^5 + O(q^6) ]
sage: eps = DirichletGroup(13).0^2 sage: ModularForms(eps,2).eisenstein_series() [ -7/13*zeta6 - 11/13 + q + (2*zeta6 + 1)*q^2 + (-3*zeta6 + 1)*q^3 + (6*zeta6 - 3)*q^4 - 4*q^5 + O(q^6), q + (zeta6 + 2)*q^2 + (-zeta6 + 3)*q^3 + (3*zeta6 + 3)*q^4 + 4*q^5 + O(q^6) ]
- eisenstein_submodule()¶
Return the Eisenstein submodule of this ambient module.
EXAMPLES:
sage: m = ModularForms(Gamma1(13),2); m Modular Forms space of dimension 13 for Congruence Subgroup Gamma1(13) of weight 2 over Rational Field sage: m.eisenstein_submodule() Eisenstein subspace of dimension 11 of Modular Forms space of dimension 13 for Congruence Subgroup Gamma1(13) of weight 2 over Rational Field
- free_module()¶
Return the free module underlying this space of modular forms.
EXAMPLES:
sage: ModularForms(37).free_module() Vector space of dimension 3 over Rational Field
- hecke_module_of_level(N)¶
Return the Hecke module of level N corresponding to self, which is the domain or codomain of a degeneracy map from self. Here N must be either a divisor or a multiple of the level of self.
EXAMPLES:
sage: ModularForms(25, 6).hecke_module_of_level(5) Modular Forms space of dimension 3 for Congruence Subgroup Gamma0(5) of weight 6 over Rational Field sage: ModularForms(Gamma1(4), 3).hecke_module_of_level(8) Modular Forms space of dimension 7 for Congruence Subgroup Gamma1(8) of weight 3 over Rational Field sage: ModularForms(Gamma1(4), 3).hecke_module_of_level(9) Traceback (most recent call last): ... ValueError: N (=9) must be a divisor or a multiple of the level of self (=4)
- hecke_polynomial(n, var='x')¶
Compute the characteristic polynomial of the Hecke operator T_n acting on this space. Except in level 1, this is computed via modular symbols, and in particular is faster to compute than the matrix itself.
EXAMPLES:
sage: ModularForms(17,4).hecke_polynomial(2) x^6 - 16*x^5 + 18*x^4 + 608*x^3 - 1371*x^2 - 4968*x + 7776
Check that this gives the same answer as computing the actual Hecke matrix (which is generally slower):
sage: ModularForms(17,4).hecke_matrix(2).charpoly() x^6 - 16*x^5 + 18*x^4 + 608*x^3 - 1371*x^2 - 4968*x + 7776
- is_ambient()¶
Return True if this an ambient space of modular forms.
This is an ambient space, so this function always returns True.
EXAMPLES:
sage: ModularForms(11).is_ambient() True sage: CuspForms(11).is_ambient() False
- modular_symbols(sign=0)¶
Return the corresponding space of modular symbols with the given sign.
EXAMPLES:
sage: S = ModularForms(11,2) sage: S.modular_symbols() Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field sage: S.modular_symbols(sign=1) Modular Symbols space of dimension 2 for Gamma_0(11) of weight 2 with sign 1 over Rational Field sage: S.modular_symbols(sign=-1) Modular Symbols space of dimension 1 for Gamma_0(11) of weight 2 with sign -1 over Rational Field
sage: ModularForms(1,12).modular_symbols() Modular Symbols space of dimension 3 for Gamma_0(1) of weight 12 with sign 0 over Rational Field
- module()¶
Return the underlying free module corresponding to this space of modular forms.
EXAMPLES:
sage: m = ModularForms(Gamma1(13),10) sage: m.free_module() Vector space of dimension 69 over Rational Field sage: ModularForms(Gamma1(13),4, GF(49,'b')).free_module() Vector space of dimension 27 over Finite Field in b of size 7^2
- new_submodule(p=None)¶
Return the new or \(p\)-new submodule of this ambient module.
INPUT:
p- (default: None), if specified return only the \(p\)-new submodule.
EXAMPLES:
sage: m = ModularForms(Gamma0(33),2); m Modular Forms space of dimension 6 for Congruence Subgroup Gamma0(33) of weight 2 over Rational Field sage: m.new_submodule() Modular Forms subspace of dimension 1 of Modular Forms space of dimension 6 for Congruence Subgroup Gamma0(33) of weight 2 over Rational Field
Another example:
sage: M = ModularForms(17,4) sage: N = M.new_subspace(); N Modular Forms subspace of dimension 4 of Modular Forms space of dimension 6 for Congruence Subgroup Gamma0(17) of weight 4 over Rational Field sage: N.basis() [ q + 2*q^5 + O(q^6), q^2 - 3/2*q^5 + O(q^6), q^3 + O(q^6), q^4 - 1/2*q^5 + O(q^6) ]
sage: ModularForms(12,4).new_submodule() Modular Forms subspace of dimension 1 of Modular Forms space of dimension 9 for Congruence Subgroup Gamma0(12) of weight 4 over Rational Field
Unfortunately (TODO) - \(p\)-new submodules aren’t yet implemented:
sage: m.new_submodule(3) # not implemented Traceback (most recent call last): ... NotImplementedError sage: m.new_submodule(11) # not implemented Traceback (most recent call last): ... NotImplementedError
- prec(new_prec=None)¶
Set or get default initial precision for printing modular forms.
INPUT:
new_prec- positive integer (default: None)
OUTPUT: if new_prec is None, returns the current precision.
EXAMPLES:
sage: M = ModularForms(1,12, prec=3) sage: M.prec() 3
sage: M.basis() [ q - 24*q^2 + O(q^3), 1 + 65520/691*q + 134250480/691*q^2 + O(q^3) ]
sage: M.prec(5) 5 sage: M.basis() [ q - 24*q^2 + 252*q^3 - 1472*q^4 + O(q^5), 1 + 65520/691*q + 134250480/691*q^2 + 11606736960/691*q^3 + 274945048560/691*q^4 + O(q^5) ]
- rank()¶
This is a synonym for
self.dimension().EXAMPLES:
sage: m = ModularForms(Gamma0(20),4) sage: m.rank() 12 sage: m.dimension() 12
- set_precision(n)¶
Set the default precision for displaying elements of this space.
EXAMPLES:
sage: m = ModularForms(Gamma1(5),2) sage: m.set_precision(10) sage: m.basis() [ 1 + 60*q^3 - 120*q^4 + 240*q^5 - 300*q^6 + 300*q^7 - 180*q^9 + O(q^10), q + 6*q^3 - 9*q^4 + 27*q^5 - 28*q^6 + 30*q^7 - 11*q^9 + O(q^10), q^2 - 4*q^3 + 12*q^4 - 22*q^5 + 30*q^6 - 24*q^7 + 5*q^8 + 18*q^9 + O(q^10) ] sage: m.set_precision(5) sage: m.basis() [ 1 + 60*q^3 - 120*q^4 + O(q^5), q + 6*q^3 - 9*q^4 + O(q^5), q^2 - 4*q^3 + 12*q^4 + O(q^5) ]