42.10 The Eisenstein Subspace

Module: sage.modular.modform.eisenstein_submodule

The Eisenstein Subspace

Module-level Functions

cyclotomic_restriction( L, K)

Given two cyclotomic fields L and K, compute the compositum M of K and L, and return a function and the index [M:K]. The function is a map that acts as follows (here M = Q($ \zeta_m$ )):

Input: element alpha in l Output: a polynomial f(x) in k[x] such that f(zeta_m) = alpha, where we view alpha as living in M. (Note that $ \zeta_m$ generates M, not L.)

sage: L = CyclotomicField(12) ; N = CyclotomicField(33) ; M = CyclotomicField(132)
sage: z, n = sage.modular.modform.eisenstein_submodule.cyclotomic_restriction(L,N)
sage: n
2

sage: z(L.0)
(-zeta33^19)*x
sage: z(L.0)(M.0)
zeta132^11

sage: z(L.0^3-L.0+1)
(zeta33^19 + zeta33^8)*x + 1
sage: z(L.0^3-L.0+1)(M.0)
zeta132^33 - zeta132^11 + 1
sage: z(L.0^3-L.0+1)(M.0) - M(L.0^3-L.0+1)
0

cyclotomic_restriction_tower( L, K)

Suppose L/K is an extension of cyclotomic fields and L=Q(zeta_m). This function computes a map with the following property:

Input: an element alpha in l Output: a polynomial f(x) in k[x] such that f(zeta_m) = alpha.

sage: L = CyclotomicField(12) ; K = CyclotomicField(6)
sage: z = sage.modular.modform.eisenstein_submodule.cyclotomic_restriction_tower(L,K)
sage: z(L.0)
x
sage: z(L.0^2+L.0)
x + zeta6

Class: EisensteinSubmodule

class EisensteinSubmodule
The Eisenstein submodule of an ambient space of modular forms.
EisensteinSubmodule( self, ambient_space)

Return the Eisenstein submodule of the given space.

sage: E = ModularForms(23,4).eisenstein_subspace() ## indirect doctest
sage: E
Eisenstein subspace of dimension 2 of Modular Forms space of dimension 7
for Congruence Subgroup Gamma0(23) of weight 4 over Rational Field
sage: E == loads(dumps(E))
True

Functions: eisenstein_submodule,$ \,$ modular_symbols

eisenstein_submodule( self)

Return the Eisenstein submodule of self. (Yes, this is just self.)

sage: E = ModularForms(23,4).eisenstein_subspace()
sage: E == E.eisenstein_submodule()
True

modular_symbols( self, [sign=0])

Return the corresponding space of modular symbols with given sign.

WARNING: If sign != 0, then the space of modular symbols will, in general, only correspond to a subspace of this space of modular forms. This can be the case for both sign +1 or -1.

sage: E = ModularForms(11,2).eisenstein_submodule()
sage: M = E.modular_symbols(); M
Modular Symbols subspace of dimension 1 of Modular Symbols space
of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field
sage: M.sign()
0

sage: M = E.modular_symbols(sign=-1); M
Modular Symbols subspace of dimension 0 of Modular Symbols space of
dimension 1 for Gamma_0(11) of weight 2 with sign -1 over Rational Field

sage: E = ModularForms(1,12).eisenstein_submodule()
sage: E.modular_symbols()
Modular Symbols subspace of dimension 1 of Modular Symbols space of
dimension 3 for Gamma_0(1) of weight 12 with sign 0 over Rational Field

sage: eps = DirichletGroup(13).0
sage: E = EisensteinForms(eps^2, 2)
sage: E.modular_symbols()
Modular Symbols subspace of dimension 2 of Modular Symbols space of
dimension 4 and level 13, weight 2, character [zeta6], sign 0, over
Cyclotomic Field of order 6 and degree 2

Special Functions: __init__,$ \,$ _repr_

_repr_( self)

Return the string representation of self.

sage: E = ModularForms(23,4).eisenstein_subspace() ## indirect doctest
sage: E._repr_()
'Eisenstein subspace of dimension 2 of Modular Forms space of dimension 7
for Congruence Subgroup Gamma0(23) of weight 4 over Rational Field'

Class: EisensteinSubmodule_eps

class EisensteinSubmodule_eps
Space of Eisenstein forms with given Dirichlet character.

sage: e = DirichletGroup(27,CyclotomicField(3)).0**2
sage: M = ModularForms(e,2,prec=10).eisenstein_subspace()
sage: M.dimension()
6

sage: M.eisenstein_series()
[
-1/3*zeta6 - 1/3 + q + (2*zeta6 - 1)*q^2 + q^3 + (-2*zeta6 - 1)*q^4 +
(-5*zeta6 + 1)*q^5 + O(q^6),
-1/3*zeta6 - 1/3 + q^3 + O(q^6),
q + (-2*zeta6 + 1)*q^2 + (-2*zeta6 - 1)*q^4 + (5*zeta6 - 1)*q^5 + O(q^6),
q + (zeta6 + 1)*q^2 + 3*q^3 + (zeta6 + 2)*q^4 + (-zeta6 + 5)*q^5 + O(q^6),
q^3 + O(q^6),
q + (-zeta6 - 1)*q^2 + (zeta6 + 2)*q^4 + (zeta6 - 5)*q^5 + O(q^6)
]
sage: M.eisenstein_subspace().T(2).matrix().fcp()
(x + zeta3 + 2) * (x + 2*zeta3 + 1) * (x - 2*zeta3 - 1)^2 * (x - zeta3 -
2)^2
sage: ModularSymbols(e,2).eisenstein_subspace().T(2).matrix().fcp()
(x + zeta3 + 2) * (x + 2*zeta3 + 1) * (x - 2*zeta3 - 1)^2 * (x - zeta3 -
2)^2

sage: M.basis()
       [
       1 - 3*zeta3*q^6 + (-2*zeta3 + 2)*q^9 + O(q^10),
       q + (5*zeta3 + 5)*q^7 + O(q^10),
       q^2 - 2*zeta3*q^8 + O(q^10),
       q^3 + (zeta3 + 2)*q^6 + 3*q^9 + O(q^10),
       q^4 - 2*zeta3*q^7 + O(q^10),
       q^5 + (zeta3 + 1)*q^8 + O(q^10)
       ]

Class: EisensteinSubmodule_g0_Q

class EisensteinSubmodule_g0_Q
Space of Eisenstein forms for Gamma0(N).

Class: EisensteinSubmodule_g1_Q

class EisensteinSubmodule_g1_Q
Space of Eisenstein forms for Gamma1(N).

Special Functions: _parameters_character

_parameters_character( self)

Return the character defining self. Since self is a space of Eisenstein forms on Gamma1(N), the character is the trivial one, which we represent by the level.

sage: EisensteinForms(Gamma1(7),4)._parameters_character()
7

Class: EisensteinSubmodule_params

class EisensteinSubmodule_params

Functions: change_ring,$ \,$ eisenstein_series,$ \,$ parameters

change_ring( self, base_ring)

Return self as a module over base_ring.

sage: E = EisensteinForms(12,2) ; E
Eisenstein subspace of dimension 5 of Modular Forms space of dimension 5
for Congruence Subgroup Gamma0(12) of weight 2 over Rational Field
sage: E.basis() 
[
1 + O(q^6),
q + 6*q^5 + O(q^6),
q^2 + O(q^6),
q^3 + O(q^6),
q^4 + O(q^6)
]
sage: E.change_ring(GF(5))
Eisenstein subspace of dimension 5 of Modular Forms space of dimension 5
for Congruence Subgroup Gamma0(12) of weight 2 over Finite Field of size 5
sage: E.change_ring(GF(5)).basis()
[
1 + O(q^6),
q + q^5 + O(q^6),
q^2 + O(q^6),
q^3 + O(q^6),
q^4 + O(q^6)
]

eisenstein_series( self)

Return the Eisenstein series that span this space (over the algebraic closure).

sage: EisensteinForms(11,2).eisenstein_series()
[
5/12 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + O(q^6)
]
sage: EisensteinForms(1,4).eisenstein_series()
[
1/240 + q + 9*q^2 + 28*q^3 + 73*q^4 + 126*q^5 + O(q^6)
]
sage: EisensteinForms(1,24).eisenstein_series()
[
236364091/131040 + q + 8388609*q^2 + 94143178828*q^3 + 70368752566273*q^4 +
11920928955078126*q^5 + O(q^6)
]
sage: EisensteinForms(5,4).eisenstein_series()
[
1/240 + q + 9*q^2 + 28*q^3 + 73*q^4 + 126*q^5 + O(q^6),
1/240 + q^5 + O(q^6)
]
sage: EisensteinForms(13,2).eisenstein_series()
[
1/2 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + O(q^6)
]

sage: E = EisensteinForms(Gamma1(7),2)
sage: E.set_precision(4)
sage: E.eisenstein_series()
[
1/4 + q + 3*q^2 + 4*q^3 + O(q^4),
1/7*zeta6 - 3/7 + q + (-2*zeta6 + 1)*q^2 + (3*zeta6 - 2)*q^3 + O(q^4),
q + (-zeta6 + 2)*q^2 + (zeta6 + 2)*q^3 + O(q^4),
-1/7*zeta6 - 2/7 + q + (2*zeta6 - 1)*q^2 + (-3*zeta6 + 1)*q^3 + O(q^4),
q + (zeta6 + 1)*q^2 + (-zeta6 + 3)*q^3 + O(q^4)
]

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)
]

sage: M = ModularForms(19,3).eisenstein_subspace()
sage: M.eisenstein_series()	    
       [
       ]

parameters( self)

Return a list of parameters for each Eisenstein series spanning self. That is, for each such series, return a triple of the form ($ \psi$ , $ \chi$ , level), where $ \psi$ and $ \chi$ are the characters defining the Eisenstein series, and level is the smallest level at which this series occurs.

sage: ModularForms(24,2).eisenstein_submodule().parameters()
[([1, 1, 1], [1, 1, 1], 2),
([1, 1, 1], [1, 1, 1], 3),
([1, 1, 1], [1, 1, 1], 4),
([1, 1, 1], [1, 1, 1], 6),
([1, 1, 1], [1, 1, 1], 8),
([1, 1, 1], [1, 1, 1], 12),
([1, 1, 1], [1, 1, 1], 24)]
sage: EisensteinForms(12,6).parameters()
[([1, 1], [1, 1], 1),
([1, 1], [1, 1], 2),
([1, 1], [1, 1], 3),
([1, 1], [1, 1], 4),
([1, 1], [1, 1], 6),
([1, 1], [1, 1], 12)]
sage: ModularForms(DirichletGroup(24).0,3).eisenstein_submodule().parameters()
[([1, 1, 1], [-1, 1, 1], 1),
([1, 1, 1], [-1, 1, 1], 2),
([1, 1, 1], [-1, 1, 1], 3),
([1, 1, 1], [-1, 1, 1], 6),
([-1, 1, 1], [1, 1, 1], 1),
([-1, 1, 1], [1, 1, 1], 2),
([-1, 1, 1], [1, 1, 1], 3),
([-1, 1, 1], [1, 1, 1], 6)]

Special Functions: _compute_q_expansion_basis,$ \,$ _parameters_character,$ \,$ _q_expansion

_compute_q_expansion_basis( self, [prec=None])

Compute a q-expansion basis for self to precision prec.

sage: EisensteinForms(22,4)._compute_q_expansion_basis(6)
[1 + O(q^6),
q + 28*q^3 - 8*q^4 + 126*q^5 + O(q^6),
q^2 + 9*q^4 + O(q^6),
O(q^6)]
sage: EisensteinForms(22,4)._compute_q_expansion_basis(15)
[1 + O(q^15),
q + 28*q^3 - 8*q^4 + 126*q^5 + 344*q^7 - 72*q^8 + 757*q^9 - 224*q^12 +
2198*q^13 + O(q^15),
q^2 + 9*q^4 + 28*q^6 + 73*q^8 + 126*q^10 + 252*q^12 + 344*q^14 + O(q^15),
q^11 + O(q^15)]

_parameters_character( self)

Return the character defining self.

sage: EisensteinForms(DirichletGroup(33).1,5)._parameters_character()
[1, zeta10]

_q_expansion( self, element, prec)

Compute a q-expansion for a given element of self, expressed as a vector of coefficients for the basis vectors of self, viewing self as a subspace of the corresponding space of modular forms.

sage: E = EisensteinForms(17,4)
sage: (11*E.0 + 3*E.1).q_expansion(20)
11 + 3*q + 27*q^2 + 84*q^3 + 219*q^4 + 378*q^5 + 756*q^6 + 1032*q^7 +
1755*q^8 + 2271*q^9 + 3402*q^10 + 3996*q^11 + 6132*q^12 + 6594*q^13 +
9288*q^14 + 10584*q^15 + 14043*q^16 + 17379*q^17 + 20439*q^18 + 20580*q^19
+ O(q^20)
sage: E._q_expansion([0,0,0,0,11,3],20)
11 + 3*q + 27*q^2 + 84*q^3 + 219*q^4 + 378*q^5 + 756*q^6 + 1032*q^7 +
1755*q^8 + 2271*q^9 + 3402*q^10 + 3996*q^11 + 6132*q^12 + 6594*q^13 +
9288*q^14 + 10584*q^15 + 14043*q^16 + 17379*q^17 + 20439*q^18 + 20580*q^19
+ O(q^20)

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