# The Eisenstein Subspace¶

class sage.modular.modform.eisenstein_submodule.EisensteinSubmodule(ambient_space)

The Eisenstein submodule of an ambient space of modular forms.

eisenstein_submodule()

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

EXAMPLES:

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

modular_symbols(sign=0)

Return the corresponding space of modular symbols with given sign. This will fail in weight 1.

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.

EXAMPLES:

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

sage: E = EisensteinForms(eps, 1); E
Eisenstein subspace of dimension 1 of Modular Forms space of dimension 1, character [zeta12] and weight 1 over Cyclotomic Field of order 12 and degree 4
sage: E.modular_symbols()
Traceback (most recent call last):
...
ValueError: the weight must be at least 2

class sage.modular.modform.eisenstein_submodule.EisensteinSubmodule_eps(ambient_space)

Space of Eisenstein forms with given Dirichlet character.

EXAMPLES:

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 + 2*zeta3 + 1) * (x + zeta3 + 2) * (x - zeta3 - 2)^2 * (x - 2*zeta3 - 1)^2
sage: ModularSymbols(e,2).eisenstein_subspace().T(2).matrix().fcp()
(x + 2*zeta3 + 1) * (x + zeta3 + 2) * (x - zeta3 - 2)^2 * (x - 2*zeta3 - 1)^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 sage.modular.modform.eisenstein_submodule.EisensteinSubmodule_g0_Q(ambient_space)

Space of Eisenstein forms for $$\Gamma_0(N)$$.

class sage.modular.modform.eisenstein_submodule.EisensteinSubmodule_g1_Q(ambient_space)

Space of Eisenstein forms for $$\Gamma_1(N)$$.

class sage.modular.modform.eisenstein_submodule.EisensteinSubmodule_gH_Q(ambient_space)

Space of Eisenstein forms for $$\Gamma_H(N)$$.

class sage.modular.modform.eisenstein_submodule.EisensteinSubmodule_params(ambient_space)

Return the Eisenstein submodule of the given space.

EXAMPLES:

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
True

change_ring(base_ring)

Return self as a module over base_ring.

EXAMPLES:

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()

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

EXAMPLES:

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()
[
]

sage: M = ModularForms(DirichletGroup(13).0, 1)
sage: M.eisenstein_series()
[
-1/13*zeta12^3 + 6/13*zeta12^2 + 4/13*zeta12 + 2/13 + q + (zeta12 + 1)*q^2 + zeta12^2*q^3 + (zeta12^2 + zeta12 + 1)*q^4 + (-zeta12^3 + 1)*q^5 + O(q^6)
]

sage: M = ModularForms(GammaH(15, [4]), 4)
sage: M.eisenstein_series()
[
1/240 + q + 9*q^2 + 28*q^3 + 73*q^4 + 126*q^5 + O(q^6),
1/240 + q^3 + O(q^6),
1/240 + q^5 + O(q^6),
1/240 + O(q^6),
1 + q - 7*q^2 - 26*q^3 + 57*q^4 + q^5 + O(q^6),
1 + q^3 + O(q^6),
q + 7*q^2 + 26*q^3 + 57*q^4 + 125*q^5 + O(q^6),
q^3 + O(q^6)
]

new_eisenstein_series()

Return a list of the Eisenstein series in this space that are new.

EXAMPLE:

sage: E = EisensteinForms(25, 4)
sage: E.new_eisenstein_series()
[q + 7*zeta4*q^2 - 26*zeta4*q^3 - 57*q^4 + O(q^6),
q - 9*q^2 - 28*q^3 + 73*q^4 + O(q^6),
q - 7*zeta4*q^2 + 26*zeta4*q^3 - 57*q^4 + O(q^6)]

new_submodule(p=None)

Return the new submodule of self.

EXAMPLE:

sage: e = EisensteinForms(Gamma0(225), 2).new_submodule(); e
Modular Forms subspace of dimension 3 of Modular Forms space of dimension 42 for Congruence Subgroup Gamma0(225) of weight 2 over Rational Field
sage: e.basis()
[
q + O(q^6),
q^2 + O(q^6),
q^4 + O(q^6)
]

parameters()

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.

EXAMPLES:

sage: ModularForms(24,2).eisenstein_submodule().parameters()
[(Dirichlet character modulo 24 of conductor 1 mapping 7 |--> 1, 13 |--> 1, 17 |--> 1, Dirichlet character modulo 24 of conductor 1 mapping 7 |--> 1, 13 |--> 1, 17 |--> 1, 2),
...
Dirichlet character modulo 24 of conductor 1 mapping 7 |--> 1, 13 |--> 1, 17 |--> 1, 24)]
sage: EisensteinForms(12,6).parameters()[-1]
(Dirichlet character modulo 12 of conductor 1 mapping 7 |--> 1, 5 |--> 1, Dirichlet character modulo 12 of conductor 1 mapping 7 |--> 1, 5 |--> 1, 12)

sage: pars = ModularForms(DirichletGroup(24).0,3).eisenstein_submodule().parameters()
sage: [(x[0].values_on_gens(),x[1].values_on_gens(),x[2]) for x in pars]
[((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)]
sage: EisensteinForms(DirichletGroup(24).0,1).parameters()
[(Dirichlet character modulo 24 of conductor 1 mapping 7 |--> 1, 13 |--> 1, 17 |--> 1, Dirichlet character modulo 24 of conductor 4 mapping 7 |--> -1, 13 |--> 1, 17 |--> 1, 1), (Dirichlet character modulo 24 of conductor 1 mapping 7 |--> 1, 13 |--> 1, 17 |--> 1, Dirichlet character modulo 24 of conductor 4 mapping 7 |--> -1, 13 |--> 1, 17 |--> 1, 2), (Dirichlet character modulo 24 of conductor 1 mapping 7 |--> 1, 13 |--> 1, 17 |--> 1, Dirichlet character modulo 24 of conductor 4 mapping 7 |--> -1, 13 |--> 1, 17 |--> 1, 3), (Dirichlet character modulo 24 of conductor 1 mapping 7 |--> 1, 13 |--> 1, 17 |--> 1, Dirichlet character modulo 24 of conductor 4 mapping 7 |--> -1, 13 |--> 1, 17 |--> 1, 6)]

sage.modular.modform.eisenstein_submodule.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 $$L$$.)

EXAMPLES:

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

sage.modular.modform.eisenstein_submodule.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 $$f(zeta_m) = alpha$$.

EXAMPLES:

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


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