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Python GammaH_class.dimension_cusp_forms方法代码示例

本文整理汇总了Python中congroup_gammaH.GammaH_class.dimension_cusp_forms方法的典型用法代码示例。如果您正苦于以下问题:Python GammaH_class.dimension_cusp_forms方法的具体用法?Python GammaH_class.dimension_cusp_forms怎么用?Python GammaH_class.dimension_cusp_forms使用的例子?那么, 这里精选的方法代码示例或许可以为您提供帮助。您也可以进一步了解该方法所在congroup_gammaH.GammaH_class的用法示例。


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示例1: dimension_cusp_forms

# 需要导入模块: from congroup_gammaH import GammaH_class [as 别名]
# 或者: from congroup_gammaH.GammaH_class import dimension_cusp_forms [as 别名]
    def dimension_cusp_forms(self, k=2, eps=None, algorithm="CohenOesterle"):
        r"""
        Return the dimension of the space of cusp forms for self, or the
        dimension of the subspace corresponding to the given character if one
        is supplied.

        INPUT:

        - ``k`` - an integer (default: 2), the weight.

        - ``eps`` - either None or a Dirichlet character modulo N, where N is
          the level of this group. If this is None, then the dimension of the
          whole space is returned; otherwise, the dimension of the subspace of
          forms of character eps.

        - ``algorithm`` -- either "CohenOesterle" (the default) or "Quer". This
          specifies the method to use in the case of nontrivial character:
          either the Cohen--Oesterle formula as described in Stein's book, or
          by Moebius inversion using the subgroups GammaH (a method due to
          Jordi Quer).

        EXAMPLES:

        We compute the same dimension in two different ways ::

            sage: K = CyclotomicField(3)
            sage: eps = DirichletGroup(7*43,K).0^2
            sage: G = Gamma1(7*43)

        Via Cohen--Oesterle: ::

            sage: Gamma1(7*43).dimension_cusp_forms(2, eps)
            28

        Via Quer's method: ::

            sage: Gamma1(7*43).dimension_cusp_forms(2, eps, algorithm="Quer")
            28

        Some more examples: ::

            sage: G.<eps> = DirichletGroup(9)
            sage: [Gamma1(9).dimension_cusp_forms(k, eps) for k in [1..10]]
            [0, 0, 1, 0, 3, 0, 5, 0, 7, 0]
            sage: [Gamma1(9).dimension_cusp_forms(k, eps^2) for k in [1..10]]
            [0, 0, 0, 2, 0, 4, 0, 6, 0, 8]
        """

        from all import Gamma0

        # first deal with special cases

        if eps is None:
            return GammaH_class.dimension_cusp_forms(self, k)

        N = self.level()
        if eps.base_ring().characteristic() != 0:
            raise ValueError

        eps = DirichletGroup(N, eps.base_ring())(eps)

        if eps.is_trivial():
            return Gamma0(N).dimension_cusp_forms(k)

        if (k <= 0) or ((k % 2) == 1 and eps.is_even()) or ((k%2) == 0 and eps.is_odd()):
            return ZZ(0)

        if k == 1:
            try:
                n = self.dimension_cusp_forms(1)
                if n == 0:
                    return ZZ(0)
                else: # never happens at present
                    raise NotImplementedError, "Computations of dimensions of spaces of weight 1 cusp forms not implemented at present"
            except NotImplementedError:
                raise

        # now the main part

        if algorithm == "Quer":
            n = eps.order()
            dim = ZZ(0)
            for d in n.divisors():
                G = GammaH_constructor(N,(eps**d).kernel())
                dim = dim + moebius(d)*G.dimension_cusp_forms(k)
            return dim//phi(n)

        elif algorithm == "CohenOesterle":
            K = eps.base_ring()
            from sage.modular.dims import CohenOesterle
            from all import Gamma0
            return ZZ( K(Gamma0(N).index() * (k-1)/ZZ(12)) + CohenOesterle(eps,k) )

        else: #algorithm not in ["CohenOesterle", "Quer"]:
            raise ValueError, "Unrecognised algorithm in dimension_cusp_forms"
开发者ID:biasse,项目名称:sage,代码行数:97,代码来源:congroup_gamma1.py


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