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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(967, base_ring=CyclotomicField(322))
M = H._module
chi = DirichletCharacter(H, M([272]))
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pari:[g,chi] = znchar(Mod(140,967))
| Modulus: | \(967\) | |
| Conductor: | \(967\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(161\) |
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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)
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\(\chi_{967}(8,\cdot)\)
\(\chi_{967}(9,\cdot)\)
\(\chi_{967}(11,\cdot)\)
\(\chi_{967}(17,\cdot)\)
\(\chi_{967}(30,\cdot)\)
\(\chi_{967}(42,\cdot)\)
\(\chi_{967}(62,\cdot)\)
\(\chi_{967}(64,\cdot)\)
\(\chi_{967}(71,\cdot)\)
\(\chi_{967}(78,\cdot)\)
\(\chi_{967}(81,\cdot)\)
\(\chi_{967}(87,\cdot)\)
\(\chi_{967}(88,\cdot)\)
\(\chi_{967}(95,\cdot)\)
\(\chi_{967}(99,\cdot)\)
\(\chi_{967}(100,\cdot)\)
\(\chi_{967}(118,\cdot)\)
\(\chi_{967}(121,\cdot)\)
\(\chi_{967}(122,\cdot)\)
\(\chi_{967}(123,\cdot)\)
\(\chi_{967}(131,\cdot)\)
\(\chi_{967}(136,\cdot)\)
\(\chi_{967}(140,\cdot)\)
\(\chi_{967}(146,\cdot)\)
\(\chi_{967}(153,\cdot)\)
\(\chi_{967}(201,\cdot)\)
\(\chi_{967}(206,\cdot)\)
\(\chi_{967}(212,\cdot)\)
\(\chi_{967}(215,\cdot)\)
\(\chi_{967}(222,\cdot)\)
...
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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 ]
\(5\) → \(e\left(\frac{136}{161}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 967 }(140, a) \) |
\(1\) | \(1\) | \(e\left(\frac{149}{161}\right)\) | \(e\left(\frac{32}{161}\right)\) | \(e\left(\frac{137}{161}\right)\) | \(e\left(\frac{136}{161}\right)\) | \(e\left(\frac{20}{161}\right)\) | \(e\left(\frac{96}{161}\right)\) | \(e\left(\frac{125}{161}\right)\) | \(e\left(\frac{64}{161}\right)\) | \(e\left(\frac{124}{161}\right)\) | \(e\left(\frac{18}{161}\right)\) |
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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)
