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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(539, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([5,21]))
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pari:[g,chi] = znchar(Mod(101,539))
| Modulus: | \(539\) | |
| Conductor: | \(539\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(210\) |
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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)
|
\(\chi_{539}(17,\cdot)\)
\(\chi_{539}(24,\cdot)\)
\(\chi_{539}(40,\cdot)\)
\(\chi_{539}(52,\cdot)\)
\(\chi_{539}(61,\cdot)\)
\(\chi_{539}(73,\cdot)\)
\(\chi_{539}(94,\cdot)\)
\(\chi_{539}(96,\cdot)\)
\(\chi_{539}(101,\cdot)\)
\(\chi_{539}(138,\cdot)\)
\(\chi_{539}(145,\cdot)\)
\(\chi_{539}(150,\cdot)\)
\(\chi_{539}(171,\cdot)\)
\(\chi_{539}(173,\cdot)\)
\(\chi_{539}(194,\cdot)\)
\(\chi_{539}(206,\cdot)\)
\(\chi_{539}(222,\cdot)\)
\(\chi_{539}(248,\cdot)\)
\(\chi_{539}(250,\cdot)\)
\(\chi_{539}(255,\cdot)\)
\(\chi_{539}(271,\cdot)\)
\(\chi_{539}(283,\cdot)\)
\(\chi_{539}(292,\cdot)\)
\(\chi_{539}(299,\cdot)\)
\(\chi_{539}(304,\cdot)\)
\(\chi_{539}(327,\cdot)\)
\(\chi_{539}(332,\cdot)\)
\(\chi_{539}(348,\cdot)\)
\(\chi_{539}(360,\cdot)\)
\(\chi_{539}(369,\cdot)\)
...
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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 ]
\((199,442)\) → \((e\left(\frac{1}{42}\right),e\left(\frac{1}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 539 }(101, a) \) |
\(1\) | \(1\) | \(e\left(\frac{151}{210}\right)\) | \(e\left(\frac{173}{210}\right)\) | \(e\left(\frac{46}{105}\right)\) | \(e\left(\frac{19}{210}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{11}{70}\right)\) | \(e\left(\frac{68}{105}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{31}{35}\right)\) |
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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)
