Python PowerSeriesRing._first_ngens方法代码示例

# 需要导入模块: from sage.rings.power_series_ring import PowerSeriesRing [as 别名]
# 或者: from sage.rings.power_series_ring.PowerSeriesRing import _first_ngens [as 别名]
def __dimension_Sp6Z(wt):
    """
    Return the dimensions of subspaces of Siegel modular forms on $Sp(6,Z)$.

    OUTPUT
    ("Total", "Miyawaki-Type-1", "Miyawaki-Type-2 (conjectured)", "Interesting")
    Remember, Miywaki type 2 is ONLY CONJECTURED!!
    """
    if not is_even(wt):
        return (0, 0, 0, 0)
    R = PowerSeriesRing(IntegerRing(), default_prec=wt + 1, names=('x',))
    (x,) = R._first_ngens(1)
    S = PowerSeriesRing(IntegerRing(), default_prec=max(2 * wt - 1,1), names=('y',))
    (y,) = S._first_ngens(1)
    H_all = 1 / ((1 - x ** 4) * (1 - x ** 12) ** 2 * (1 - x ** 14) * (1 - x ** 18) *
                (1 - x ** 20) * (1 - x ** 30)) * (
                    1 + x ** 6 + x ** 10 + x ** 12 + 3 * x ** 16 + 2 * x ** 18 + 2 * x ** 20 +
                    5 * x ** 22 + 4 * x ** 24 + 5 * x ** 26 + 7 * x ** 28 + 6 * x ** 30 + 9 * x ** 32 +
                    10 * x ** 34 + 10 * x ** 36 + 12 * x ** 38 + 14 * x ** 40 + 15 * x ** 42 + 16 * x ** 44 +
                    18 * x ** 46 + 18 * x ** 48 + 19 * x ** 50 + 21 * x ** 52 + 19 * x ** 54 + 21 * x ** 56 +
                    21 * x ** 58 + 19 * x ** 60 + 21 * x ** 62 + 19 * x ** 64 + 18 * x ** 66 + 18 * x ** 68 +
                    16 * x ** 70 + 15 * x ** 72 + 14 * x ** 74 + 12 * x ** 76 + 10 * x ** 78 + 10 * x ** 80 +
                    9 * x ** 82 + 6 * x ** 84 + 7 * x ** 86 + 5 * x ** 88 + 4 * x ** 90 + 5 * x ** 92 +
                    2 * x ** 94 + 2 * x ** 96 + 3 * x ** 98 + x ** 102 + x ** 104 + x ** 108 + x ** 114)

    H_noncusp = 1 / (1 - x ** 4) / (1 - x ** 6) / (1 - x ** 10) / (1 - x ** 12)
    H_E = y ** 12 / (1 - y ** 4) / (1 - y ** 6)
    H_Miyawaki1 = H_E[wt] * H_E[2 * wt - 4]
    H_Miyawaki2 = H_E[wt - 2] * H_E[2 * wt - 2]
    a, b, c, d = H_all[wt], H_noncusp[wt], H_Miyawaki1, H_Miyawaki2
    return (a, c, d, a - b - c - d)

# 需要导入模块: from sage.rings.power_series_ring import PowerSeriesRing [as 别名]
# 或者: from sage.rings.power_series_ring.PowerSeriesRing import _first_ngens [as 别名]
def _dimension_Sp4Z( wt_range):
    """
    Return the dimensions of subspaces of Siegel modular forms on $Sp(4,Z)$.

    OUTPUT
        ("Total", "Eisenstein", "Klingen", "Maass", "Interesting")
    """
    headers = ['Total', 'Eisenstein', 'Klingen', 'Maass', 'Interesting']

    R = PowerSeriesRing( IntegerRing(), default_prec = wt_range[-1] + 1, names = ('x',))
    (x,) = R._first_ngens(1)
    H_all = 1 / (1 - x ** 4) / (1 - x ** 6) / (1 - x ** 10) / (1 - x ** 12)
    H_Kl = x ** 12 / (1 - x ** 4) / (1 - x ** 6)
    H_MS = (x ** 10 + x ** 12) / (1 - x ** 4) / (1 - x ** 6)

    dct = dict( (k,
                 { 'Total': H_all[k], 
                   'Eisenstein': 1 if k >= 4 else 0,
                   'Klingen': H_Kl[k],
                   'Maass': H_MS[k],
                   'Interesting': H_all[k]-(1 if k >= 4 else 0)-H_Kl[k]-H_MS[k]
                   }
                 if is_even(k) else
                 { 'Total': H_all[k-35], 
                   'Eisenstein': 0,
                   'Klingen': 0,
                   'Maass': 0,
                   'Interesting': H_all[k-35]
                   }
                 ) for k in wt_range)

    return headers, dct

# 需要导入模块: from sage.rings.power_series_ring import PowerSeriesRing [as 别名]
# 或者: from sage.rings.power_series_ring.PowerSeriesRing import _first_ngens [as 别名]
def _dimension_Gamma0_4_half(k):
    """
    Return the dimensions of subspaces of Siegel modular forms$Gamma0(4)$
    of half integral weight  k - 1/2.

    INPUT
        The realweight is k-1/2

    OUTPUT
        ('Total', 'Non cusp', 'Cusp')

    REMARK
        Note that formula from Hayashida's and Ibukiyama's paper has formula
        that coefficient of x^w is for weight (w+1/2). So here w=k-1.
    """
    R = PowerSeriesRing(IntegerRing(), default_prec=k, names=("x",))
    (x,) = R._first_ngens(1)
    H_all = 1 / (1 - x) / (1 - x ** 2) ** 2 / (1 - x ** 3)
    H_cusp = (
        (2 * x ** 5 + x ** 7 + x ** 9 - 2 * x ** 11 + 4 * x ** 6 - x ** 8 + x ** 10 - 3 * x ** 12 + x ** 14)
        / (1 - x ** 2) ** 2
        / (1 - x ** 6)
    )
    a, c = H_all[k - 1], H_cusp[k - 1]
    return (a, a - c, c)

# 需要导入模块: from sage.rings.power_series_ring import PowerSeriesRing [as 别名]
# 或者: from sage.rings.power_series_ring.PowerSeriesRing import _first_ngens [as 别名]
def _dimension_Gamma0_4(wt):
    """
    Return the dimensions of subspaces of Siegel modular forms on $Gamma0(4)$.

    OUTPUT
        ( "Total",)

    REMARK
        Not completely implemented
    """
    R = PowerSeriesRing(IntegerRing(), default_prec=wt + 1, names=("x",))
    (x,) = R._first_ngens(1)
    H_all = (1 + x ** 4)(1 + x ** 11) / (1 - x ** 2) ** 3 / (1 - x ** 6)
    return (H_all[wt],)

# 需要导入模块: from sage.rings.power_series_ring import PowerSeriesRing [as 别名]
# 或者: from sage.rings.power_series_ring.PowerSeriesRing import _first_ngens [as 别名]
def _dimension_Gamma0_3(wt):
    """
    Return the dimensions of subspaces of Siegel modular forms on $Gamma0(3)$.

    OUTPUT
        ( "Total")

    REMARK
        Only total dimension implemented.
    """
    R = PowerSeriesRing( IntegerRing(), default_prec = wt + 1, names=('x',))
    (x,) = R._first_ngens(1)
    H_all = (1 + 2 * x ** 4 + x ** 6 + x ** 15 * (1 + 2 * x ** 2 + x ** 6)) / (1 - x ** 2) / (1 - x ** 4) / (1 - x ** 6) ** 2
    return ( H_all[wt],)

# 需要导入模块: from sage.rings.power_series_ring import PowerSeriesRing [as 别名]
# 或者: from sage.rings.power_series_ring.PowerSeriesRing import _first_ngens [as 别名]
def _dimension_Gamma0_4_psi_4(wt):
    """
    Return the dimensions of subspaces of Siegel modular forms
    on $Gamma_0(4)$
    with character $\psi_4$.

    OUTPUT
        ( "Total")

    REMARK
        The formula for odd weights is unknown or not obvious from the paper.
    """
    R = PowerSeriesRing(IntegerRing(), default_prec=wt + 1, names=("x",))
    (x,) = R._first_ngens(1)
    H_all_even = (x ** 12 + x ** 14) / (1 - x ** 2) ** 3 / (1 - x ** 6)
    if is_even(wt):
        return (H_all_even[wt],)
    else:
        raise NotImplementedError("Dimensions of $M_{k}(\Gamma_0(4), \psi_4)$ for odd $k$ not implemented")

# 需要导入模块: from sage.rings.power_series_ring import PowerSeriesRing [as 别名]
# 或者: from sage.rings.power_series_ring.PowerSeriesRing import _first_ngens [as 别名]
def _dimension_Gamma0_3_psi_3(wt):
    """
    Return the dimensions of the space of Siegel modular forms
    on $Gamma_0(3)$ with character $\psi_3$.

    OUTPUT
        ( "Total")

    REMARK
        Not completely implemented
    """
    R = PowerSeriesRing(IntegerRing(), default_prec=wt + 1, names=("x",))
    (x,) = R._first_ngens(1)
    B = 1 / (1 - x ** 1) / (1 - x ** 3) / (1 - x ** 4) / (1 - x ** 3)
    H_all_odd = B
    H_all_even = B * x ** 14
    if is_even(wt):
        return (H_all_even[wt],)
    else:
        return (H_all_odd[wt],)