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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(289, base_ring=CyclotomicField(68))
M = H._module
chi = DirichletCharacter(H, M([43]))
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pari:[g,chi] = znchar(Mod(123,289))
| Modulus: | \(289\) | |
| Conductor: | \(289\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(68\) |
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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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\(\chi_{289}(4,\cdot)\)
\(\chi_{289}(13,\cdot)\)
\(\chi_{289}(21,\cdot)\)
\(\chi_{289}(30,\cdot)\)
\(\chi_{289}(47,\cdot)\)
\(\chi_{289}(55,\cdot)\)
\(\chi_{289}(64,\cdot)\)
\(\chi_{289}(72,\cdot)\)
\(\chi_{289}(81,\cdot)\)
\(\chi_{289}(89,\cdot)\)
\(\chi_{289}(98,\cdot)\)
\(\chi_{289}(106,\cdot)\)
\(\chi_{289}(115,\cdot)\)
\(\chi_{289}(123,\cdot)\)
\(\chi_{289}(132,\cdot)\)
\(\chi_{289}(140,\cdot)\)
\(\chi_{289}(149,\cdot)\)
\(\chi_{289}(157,\cdot)\)
\(\chi_{289}(166,\cdot)\)
\(\chi_{289}(174,\cdot)\)
\(\chi_{289}(183,\cdot)\)
\(\chi_{289}(191,\cdot)\)
\(\chi_{289}(200,\cdot)\)
\(\chi_{289}(208,\cdot)\)
\(\chi_{289}(217,\cdot)\)
\(\chi_{289}(225,\cdot)\)
\(\chi_{289}(234,\cdot)\)
\(\chi_{289}(242,\cdot)\)
\(\chi_{289}(259,\cdot)\)
\(\chi_{289}(268,\cdot)\)
...
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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 ]
\(3\) → \(e\left(\frac{43}{68}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 289 }(123, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{43}{68}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{53}{68}\right)\) | \(e\left(\frac{1}{68}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{65}{68}\right)\) | \(e\left(\frac{37}{68}\right)\) |
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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)
