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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(889, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([42,22]))
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pari:[g,chi] = znchar(Mod(324,889))
| Modulus: | \(889\) | |
| Conductor: | \(889\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(63\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
|
| Real: | no |
| Primitive: | yes |
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sage:chi.is_primitive()
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pari:#znconreyconductor(g,chi)==1
|
| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
|
\(\chi_{889}(11,\cdot)\)
\(\chi_{889}(18,\cdot)\)
\(\chi_{889}(30,\cdot)\)
\(\chi_{889}(74,\cdot)\)
\(\chi_{889}(79,\cdot)\)
\(\chi_{889}(88,\cdot)\)
\(\chi_{889}(121,\cdot)\)
\(\chi_{889}(142,\cdot)\)
\(\chi_{889}(144,\cdot)\)
\(\chi_{889}(198,\cdot)\)
\(\chi_{889}(240,\cdot)\)
\(\chi_{889}(247,\cdot)\)
\(\chi_{889}(263,\cdot)\)
\(\chi_{889}(289,\cdot)\)
\(\chi_{889}(298,\cdot)\)
\(\chi_{889}(324,\cdot)\)
\(\chi_{889}(326,\cdot)\)
\(\chi_{889}(394,\cdot)\)
\(\chi_{889}(417,\cdot)\)
\(\chi_{889}(443,\cdot)\)
\(\chi_{889}(485,\cdot)\)
\(\chi_{889}(529,\cdot)\)
\(\chi_{889}(534,\cdot)\)
\(\chi_{889}(550,\cdot)\)
\(\chi_{889}(557,\cdot)\)
\(\chi_{889}(592,\cdot)\)
\(\chi_{889}(606,\cdot)\)
\(\chi_{889}(632,\cdot)\)
\(\chi_{889}(669,\cdot)\)
\(\chi_{889}(676,\cdot)\)
...
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sage:chi.galois_orbit()
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pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((255,638)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{11}{63}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 889 }(324, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{62}{63}\right)\) |
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sage:chi.jacobi_sum(n)
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sage:chi.gauss_sum(a)
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pari:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
