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sha:~ was$ sage
--------------------------------------------------------
| SAGE Version 1.2.3, Build Date: 2006-04-20-2054      |
| Distributed under the GNU General Public License V2. |
| For help type <anything>? or <anything>??.           |
--------------------------------------------------------

  

sage: 
sage: 
sage: DirichletGroup?
Type:           function
Base Class:     <type 'function'>
String Form:    <function DirichletGroup at 0x9e06c30>
Namespace:      Interactive
File:           /Users/was/s/local/lib/python2.4/site-packages/sage/modular/dirichlet.py
Definition:     DirichletGroup(modulus, base_ring=None, zeta=None, zeta_order=None)
Docstring:
    <no docstring>

sage: G = DirichletGroup(11)
sage: G
 _3 = Group of Dirichlet characters of modulus 11 over Cyclotomic Field of order 10 and degree 4
sage: G.list()
 _4 = 
[[1],
 [zeta10],
 [zeta10^2],
 [zeta10^3],
 [zeta10^3 - zeta10^2 + zeta10 - 1],
 [-1],
 [-zeta10],
 [-zeta10^2],
 [-zeta10^3],
 [-zeta10^3 + zeta10^2 - zeta10 + 1]]
sage: G.unit_gens()
 _5 = [2]
sage: eps = G.0
sage: eps
 _7 = [zeta10]
sage: eps.order()
 _8 = 10
sage: eps(3)
 _9 = -zeta10^3
sage: eps(7)
_10 = -zeta10^2
sage: eps(11)
_11 = 0
sage: eps(2006)
_12 = zeta10^2
sage: eps.va
eps.values          eps.values_on_gens  
sage: eps.values()
_13 = 
[0,
 1,
 zeta10,
 -zeta10^3,
 zeta10^2,
 zeta10^3 - zeta10^2 + zeta10 - 1,
 -zeta10^3 + zeta10^2 - zeta10 + 1,
 -zeta10^2,
 zeta10^3,
 -zeta10,
 -1]
sage: eps.
eps.Hom
eps._DirichletCharacter__eval_at_minus_one
eps._DirichletCharacter__is_trivial
eps._DirichletCharacter__modulus
eps._DirichletCharacter__order
eps._DirichletCharacter__values
eps._DirichletCharacter__values_on_gens
eps._add_
eps._coeff_repr
eps._div_
eps._gap_
eps._gap_init_
eps._gp_
eps._gp_init_
eps._im_gens_
eps._interface_
eps._interface_is_cached_
eps._is_atomic
eps._kash_
eps._kash_init_
eps._latex_coeff_repr
eps._macaulay2_
eps._macaulay2_init_
eps._magma_
eps._magma_init_
eps._maple_
eps._maple_init_
eps._mathematica_
eps._mathematica_init_
eps._maxima_
eps._maxima_init_
eps._mul_
eps._octave_
eps._octave_init_
eps._parent
eps._pari_
eps._pari_init_
eps._plot_
eps._repr_
eps._rich_to_bool
eps._sage_
eps._set_parent
eps._singular_
eps._singular_init_
eps.bar
eps.base_ring
eps.bernoulli
eps.category
sage: e = eps.bar()
sage: e
_15 = [-zeta10^3 + zeta10^2 - zeta10 + 1]
sage: eps
_16 = [zeta10]
sage: e*eps
_17 = [1]
sage: eps.modulus()
_18 = 11
sage: t = e*eps
sage: t
_20 = [1]
sage: t.modulus()
_21 = 11
sage: k = GF(25)
sage: G = DirichletGroup(25, k)
sage: G
_24 = Group of Dirichlet characters of modulus 25 over Finite Field in a of size 5^2
sage: len(G)
_25 = 4
sage: G.list()
_26 = [[1], [2], [4], [3]]
sage: v = [p for p in primes(300) if (p-1)%20 == 0]
sage: v
_28 = [41, 61, 101, 181, 241, 281]
sage: G = DirichletGroup(25, GF(41))
sage: G
_30 = Group of Dirichlet characters of modulus 25 over Finite Field of size 41
sage: G.unit_gens()
_31 = [2]
sage: len(G)
_32 = 20
sage: G.list()
_33 = 
[[1],
 [36],
 [25],
 [39],
 [10],
 [32],
 [4],
 [21],
 [18],
 [33],
 [40],
 [5],
 [16],
 [2],
 [31],
 [9],
 [37],
 [20],
 [23],
 [8]]
sage: e = G.0
sage: e
_35 = [36]
sage: e(3)
_36 = 21
sage: factor(20060
   ....: )
_37 = 2^2 * 5 * 17 * 59
sage: factor(2006) 
_38 = 2 * 17 * 59
sage: G = DirichletGroup(2006)
sage: G.order()
_40 = 928
sage: G.unit_gens()
_41 = [1, 1771, 1123]
sage: e = G.0
sage: e
_43 = [1, 1, 1]
sage: G.1
_44 = [1, zeta464^29, 1]
sage: G.2
_45 = [1, 1, zeta464^8]
sage: G.0
^P_46 = [1, 1, 1]
sage: G.1
_47 = [1, zeta464^29, 1]
sage: G.2
_48 = [1, 1, zeta464^8]
sage: (G.0).conductor()
_49 = 1
sage: G.1 
_50 = [1, zeta464^29, 1]
sage: (G.1).conductor()
_51 = 17
sage: (G.2).conductor()
_52 = 59
sage: e = G.1^3 * G.2^5 
sage: e
_54 = [1, zeta464^87, zeta464^40]
sage: e.conductor()
_55 = 1003
sage: e.is_primitive()
_56 = False
sage: e.modulus()
_57 = 2006
sage: e.
e.Hom
e._DirichletCharacter__conductor
e._DirichletCharacter__decomp
e._DirichletCharacter__eval_at_minus_one
e._DirichletCharacter__is_primitive
e._DirichletCharacter__is_trivial
e._DirichletCharacter__modulus
e._DirichletCharacter__values_on_gens
e._add_
e._coeff_repr
e._div_
e._gap_
e._gap_init_
e._gp_
e._gp_init_
e._im_gens_
e._interface_
e._interface_is_cached_
e._is_atomic
e._kash_
e._kash_init_
e._latex_coeff_repr
e._macaulay2_
e._macaulay2_init_
e._magma_
e._magma_init_
e._maple_
e._maple_init_
e._mathematica_
e._mathematica_init_
e._maxima_
e._maxima_init_
e._mul_
e._octave_
e._octave_init_
e._parent
e._pari_
e._pari_init_
e._plot_
e._repr_
e._rich_to_bool
e._sage_
e._set_parent
e._singular_
e._singular_init_
e.bar
e.base_ring
e.bernoulli
e.category
e.change_base
e.change_base_ring
e.conductor
e.db
e.decomposition
e.dump
e.dumps
e.extend
e.galois_orbit
e.gauss_sum
e.gauss_sum_numerical
e.is_even
e.is_odd
e.is_primitive
e.is_trivial
e.maximize_base_ring
e.minimize_base_ring
e.modulus
e.multiplicative_order
e.order
e.parent
e.rename
e.reset_name
sage: e.decomposition()
_58 = [[1], [zeta464^87], [zeta464^40]]
sage: e.
e.Hom
e._DirichletCharacter__conductor
e._DirichletCharacter__decomp
e._DirichletCharacter__eval_at_minus_one
e._DirichletCharacter__is_primitive
e._DirichletCharacter__is_trivial
e._DirichletCharacter__modulus
e._DirichletCharacter__values_on_gens
e._add_
e._coeff_repr
e._div_
e._gap_
e._gap_init_
e._gp_
e._gp_init_
e._im_gens_
e._interface_
e._interface_is_cached_
e._is_atomic
e._kash_
e._kash_init_
e._latex_coeff_repr
e._macaulay2_
e._macaulay2_init_
e._magma_
e._magma_init_
e._maple_
e._maple_init_
e._mathematica_
e._mathematica_init_
e._maxima_
e._maxima_init_
e._mul_
e._octave_
e._octave_init_
e._parent
e._pari_
e._pari_init_
e._plot_
e._repr_
e._rich_to_bool
e._sage_
e._set_parent
e._singular_
e._singular_init_
e.bar
e.base_ring
e.bernoulli
e.category
e.change_base
e.change_base_ring
e.conductor
e.db
e.decomposition
e.dump
e.dumps
e.extend
e.galois_orbit
e.gauss_sum
e.gauss_sum_numerical
e.is_even
e.is_odd
e.is_primitive
e.is_trivial
e.maximize_base_ring
e.minimize_base_ring
e.modulus
e.multiplicative_order
e.order
e.parent
e.rename
e.reset_name
sage: e.gaus
e.gauss_sum            e.gauss_sum_numerical  
sage: e.gauss_sum?
sage: e.gauss_sum() 
^C------------------------------------------------------------
Traceback (most recent call last):
  File "<ipython console>", line 1, in ?
  File "/Users/was/s/local/lib/python2.4/site-packages/sage/modular/dirichlet.py", line 424, in gauss_sum
    L = rings.CyclotomicField(arith.lcm(m,G.zeta_order()))
  File "/Users/was/s/local/lib/python2.4/site-packages/sage/rings/number_field/number_field.py", line 127, in CyclotomicField
    return NumberField_cyclotomic(n)
  File "/Users/was/s/local/lib/python2.4/site-packages/sage/rings/number_field/number_field.py", line 544, in __init__
    f = QQ['x'].cyclotomic_polynomial(n)
  File "/Users/was/s/local/lib/python2.4/site-packages/sage/rings/polynomial_ring.py", line 239, in cyclotomic_polynomial
    f = pari.polcyclo(n)
KeyboardInterrupt

sage: e.gauss_sum_numerical()
^C------------------------------------------------------------
Traceback (most recent call last):
  File "<ipython console>", line 1, in ?
  File "/Users/was/s/local/lib/python2.4/site-packages/sage/modular/dirichlet.py", line 490, in gauss_sum_numerical
    g += phi(c)*z
  File "/Users/was/s/local/lib/python2.4/site-packages/sage/rings/morphism.py", line 317, in __call__
    return Morphism.__call__(self, x)
  File "/Users/was/s/local/lib/python2.4/site-packages/sage/categories/morphism.py", line 80, in __call__
    return self._call_(y)
  File "/Users/was/s/local/lib/python2.4/site-packages/sage/rings/morphism.py", line 428, in _call_
    return x._im_gens_(self.codomain(), self.im_gens())
  File "/Users/was/s/local/lib/python2.4/site-packages/sage/rings/number_field/number_field_element.py", line 101, in _im_gens_
    return codomain(self.__element(im_gens[0]))
  File "/Users/was/s/local/lib/python2.4/site-packages/sage/rings/polynomial_element.py", line 152, in __call__
    result = result * a + self[i]
  File "/Users/was/s/local/lib/python2.4/site-packages/sage/rings/polynomial_element.py", line 1485, in __getitem__
    return QQ(self.__poly[n])
  File "gen.pyx", line 278, in gen.gen.__getitem__
KeyboardInterrupt

sage: e.
e.Hom
e._DirichletCharacter__conductor
e._DirichletCharacter__decomp
e._DirichletCharacter__eval_at_minus_one
e._DirichletCharacter__is_primitive
e._DirichletCharacter__is_trivial
e._DirichletCharacter__modulus
e._DirichletCharacter__values
e._DirichletCharacter__values_on_gens
e._add_
e._coeff_repr
e._div_
e._gap_
e._gap_init_
e._gp_
e._gp_init_
e._im_gens_
e._interface_
e._interface_is_cached_
e._is_atomic
e._kash_
e._kash_init_
e._latex_coeff_repr
e._macaulay2_
e._macaulay2_init_
e._magma_
e._magma_init_
e._maple_
e._maple_init_
e._mathematica_
e._mathematica_init_
e._maxima_
e._maxima_init_
e._mul_
e._octave_
e._octave_init_
e._parent
e._pari_
e._pari_init_
e._plot_
e._repr_
e._rich_to_bool
e._sage_
e._set_parent
e._singular_
e._singular_init_
e.bar
e.base_ring
e.bernoulli
e.category
e.change_base
e.change_base_ring
e.conductor
e.db
e.decomposition
e.dump
e.dumps
e.extend
e.galois_orbit
e.gauss_sum
e.gauss_sum_numerical
e.is_even
e.is_odd
e.is_primitive
e.is_trivial
e.maximize_base_ring
e.minimize_base_ring
e.modulus
e.multiplicative_order
e.order
e.parent
e.rename
e.reset_name
e.restrict
e.save
e.values
e.values_on_gens
e.version
sage: