The Eisenstein subspace - Modular Forms

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# The Eisenstein subspace

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

  Bases: `ModularFormsSubmodule`

  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

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

    Python

        >>> from sage.all import *
        >>> E = ModularForms(Integer(23),Integer(4)).eisenstein_subspace()
        >>> 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

        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 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

    Python

        >>> from sage.all import *
        >>> E = ModularForms(Integer(11),Integer(2)).eisenstein_submodule()
        >>> 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
        >>> M.sign()
        0

        >>> M = E.modular_symbols(sign=-Integer(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

        >>> E = ModularForms(Integer(1),Integer(12)).eisenstein_submodule()
        >>> 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

        >>> eps = DirichletGroup(Integer(13)).gen(0)
        >>> E = EisensteinForms(eps**Integer(2), Integer(2))
        >>> 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

        >>> E = EisensteinForms(eps, Integer(1)); E
        Eisenstein subspace of dimension 1 of Modular Forms space of character
         [zeta12] and weight 1 over Cyclotomic Field of order 12 and degree 4
        >>> 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)

  Bases: `EisensteinSubmodule_params`

  Space of Eisenstein forms with given Dirichlet character.

  EXAMPLES:

  Sage

      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)]

  Python

      >>> from sage.all import *
      >>> e = DirichletGroup(Integer(27),CyclotomicField(Integer(3))).gen(0)**Integer(2)
      >>> M = ModularForms(e,Integer(2),prec=Integer(10)).eisenstein_subspace()
      >>> M.dimension()
      6

      >>> 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)]
      >>> M.eisenstein_subspace().T(Integer(2)).matrix().fcp()
      (x + 2*zeta3 + 1) * (x + zeta3 + 2) * (x - zeta3 - 2)^2 * (x - 2*zeta3 - 1)^2
      >>> ModularSymbols(e,Integer(2)).eisenstein_subspace().T(Integer(2)).matrix().fcp()
      (x + 2*zeta3 + 1) * (x + zeta3 + 2) * (x - zeta3 - 2)^2 * (x - 2*zeta3 - 1)^2

      >>> 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)

  Bases: `EisensteinSubmodule_params`

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

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

  Bases: `EisensteinSubmodule_gH_Q`

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

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

  Bases: `EisensteinSubmodule_params`

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

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

  Bases: `EisensteinSubmodule`

  change_ring(base_ring)

    Return `self` as a module over `base_ring`.

    EXAMPLES:

    Sage

        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)]

    Python

        >>> from sage.all import *
        >>> E = EisensteinForms(Integer(12),Integer(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
        >>> 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)]
        >>> E.change_ring(GF(Integer(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
        >>> E.change_ring(GF(Integer(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

        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)]

    Python

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

        >>> E = EisensteinForms(Gamma1(Integer(7)),Integer(2))
        >>> E.set_precision(Integer(4))
        >>> 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)]

        >>> eps = DirichletGroup(Integer(13)).gen(0)**Integer(2)
        >>> ModularForms(eps,Integer(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)]

        >>> M = ModularForms(Integer(19),Integer(3)).eisenstein_subspace()
        >>> M.eisenstein_series()
        []

        >>> M = ModularForms(DirichletGroup(Integer(13)).gen(0), Integer(1))
        >>> 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)]

        >>> M = ModularForms(GammaH(Integer(15), [Integer(4)]), Integer(4))
        >>> 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.

    EXAMPLES:

    Sage

        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)]

    Python

        >>> from sage.all import *
        >>> E = EisensteinForms(Integer(25), Integer(4))
        >>> 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`.

    EXAMPLES:

    Sage

        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)]

    Python

        >>> from sage.all import *
        >>> e = EisensteinForms(Gamma0(Integer(225)), Integer(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
        >>> 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

        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)]

    Python

        >>> from sage.all import *
        >>> ModularForms(Integer(24),Integer(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)]
        >>> EisensteinForms(Integer(12),Integer(6)).parameters()[-Integer(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)

        >>> pars = ModularForms(DirichletGroup(Integer(24)).gen(0),Integer(3)).eisenstein_submodule().parameters()
        >>> [(x[Integer(0)].values_on_gens(),x[Integer(1)].values_on_gens(),x[Integer(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)]
        >>> EisensteinForms(DirichletGroup(Integer(24)).gen(0),Integer(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 \(f\) and the index \([M:K]\).

  The function \(f\) is a map that acts as follows (here \(M =\QQ(\zeta_m)\)):

  > 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

      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

  Python

      >>> from sage.all import *
      >>> L = CyclotomicField(Integer(12)); N = CyclotomicField(Integer(33)); M = CyclotomicField(Integer(132))
      >>> z, n = sage.modular.modform.eisenstein_submodule.cyclotomic_restriction(L, N)
      >>> n
      2

      >>> z(L.gen(0))
      -zeta33^19*x
      >>> z(L.gen(0))(M.gen(0))
      zeta132^11

      >>> z(L.gen(0)**Integer(3) - L.gen(0) + Integer(1))
      (zeta33^19 + zeta33^8)*x + 1
      >>> z(L.gen(0)**Integer(3) - L.gen(0) + Integer(1))(M.gen(0))
      zeta132^33 - zeta132^11 + 1
      >>> z(L.gen(0)**Integer(3) - L.gen(0) + Integer(1))(M.gen(0)) - M(L.gen(0)**Integer(3) - L.gen(0) + Integer(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: element alpha in \(L\) OUTPUT: a polynomial \(f(x)\) in \(K[x]\) such that \(f(\zeta_m) = alpha\)

  EXAMPLES:

  Sage

      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

  Python

      >>> from sage.all import *
      >>> L = CyclotomicField(Integer(12)) ; K = CyclotomicField(Integer(6))
      >>> z = sage.modular.modform.eisenstein_submodule.cyclotomic_restriction_tower(L,K)
      >>> z(L.gen(0))
      x
      >>> z(L.gen(0)**Integer(2)+L.gen(0))
      x + zeta6