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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(875, base_ring=CyclotomicField(300))
M = H._module
chi = DirichletCharacter(H, M([27,250]))
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pari:[g,chi] = znchar(Mod(12,875))
| Modulus: | \(875\) | |
| Conductor: | \(875\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(300\) |
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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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\(\chi_{875}(3,\cdot)\)
\(\chi_{875}(12,\cdot)\)
\(\chi_{875}(17,\cdot)\)
\(\chi_{875}(33,\cdot)\)
\(\chi_{875}(38,\cdot)\)
\(\chi_{875}(47,\cdot)\)
\(\chi_{875}(52,\cdot)\)
\(\chi_{875}(73,\cdot)\)
\(\chi_{875}(87,\cdot)\)
\(\chi_{875}(103,\cdot)\)
\(\chi_{875}(108,\cdot)\)
\(\chi_{875}(117,\cdot)\)
\(\chi_{875}(122,\cdot)\)
\(\chi_{875}(138,\cdot)\)
\(\chi_{875}(152,\cdot)\)
\(\chi_{875}(173,\cdot)\)
\(\chi_{875}(178,\cdot)\)
\(\chi_{875}(187,\cdot)\)
\(\chi_{875}(192,\cdot)\)
\(\chi_{875}(208,\cdot)\)
\(\chi_{875}(213,\cdot)\)
\(\chi_{875}(222,\cdot)\)
\(\chi_{875}(227,\cdot)\)
\(\chi_{875}(248,\cdot)\)
\(\chi_{875}(262,\cdot)\)
\(\chi_{875}(278,\cdot)\)
\(\chi_{875}(283,\cdot)\)
\(\chi_{875}(292,\cdot)\)
\(\chi_{875}(297,\cdot)\)
\(\chi_{875}(313,\cdot)\)
...
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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 ]
\((127,626)\) → \((e\left(\frac{9}{100}\right),e\left(\frac{5}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 875 }(12, a) \) |
\(1\) | \(1\) | \(e\left(\frac{227}{300}\right)\) | \(e\left(\frac{139}{300}\right)\) | \(e\left(\frac{77}{150}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{27}{100}\right)\) | \(e\left(\frac{139}{150}\right)\) | \(e\left(\frac{13}{75}\right)\) | \(e\left(\frac{293}{300}\right)\) | \(e\left(\frac{1}{100}\right)\) | \(e\left(\frac{2}{75}\right)\) |
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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)
