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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([123,100]))
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pari:[g,chi] = znchar(Mod(177,875))
| 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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{875}(2,\cdot)\)
\(\chi_{875}(23,\cdot)\)
\(\chi_{875}(37,\cdot)\)
\(\chi_{875}(53,\cdot)\)
\(\chi_{875}(58,\cdot)\)
\(\chi_{875}(67,\cdot)\)
\(\chi_{875}(72,\cdot)\)
\(\chi_{875}(88,\cdot)\)
\(\chi_{875}(102,\cdot)\)
\(\chi_{875}(123,\cdot)\)
\(\chi_{875}(128,\cdot)\)
\(\chi_{875}(137,\cdot)\)
\(\chi_{875}(142,\cdot)\)
\(\chi_{875}(158,\cdot)\)
\(\chi_{875}(163,\cdot)\)
\(\chi_{875}(172,\cdot)\)
\(\chi_{875}(177,\cdot)\)
\(\chi_{875}(198,\cdot)\)
\(\chi_{875}(212,\cdot)\)
\(\chi_{875}(228,\cdot)\)
\(\chi_{875}(233,\cdot)\)
\(\chi_{875}(242,\cdot)\)
\(\chi_{875}(247,\cdot)\)
\(\chi_{875}(263,\cdot)\)
\(\chi_{875}(277,\cdot)\)
\(\chi_{875}(298,\cdot)\)
\(\chi_{875}(303,\cdot)\)
\(\chi_{875}(312,\cdot)\)
\(\chi_{875}(317,\cdot)\)
\(\chi_{875}(333,\cdot)\)
...
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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 ]
\((127,626)\) → \((e\left(\frac{41}{100}\right),e\left(\frac{1}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 875 }(177, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{23}{300}\right)\) | \(e\left(\frac{61}{300}\right)\) | \(e\left(\frac{23}{150}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{23}{100}\right)\) | \(e\left(\frac{61}{150}\right)\) | \(e\left(\frac{37}{75}\right)\) | \(e\left(\frac{107}{300}\right)\) | \(e\left(\frac{99}{100}\right)\) | \(e\left(\frac{23}{75}\right)\) |
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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)
