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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(805, base_ring=CyclotomicField(132))
M = H._module
chi = DirichletCharacter(H, M([33,22,42]))
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pari:[g,chi] = znchar(Mod(17,805))
| Modulus: | \(805\) | |
| Conductor: | \(805\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(132\) |
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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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\(\chi_{805}(17,\cdot)\)
\(\chi_{805}(33,\cdot)\)
\(\chi_{805}(38,\cdot)\)
\(\chi_{805}(103,\cdot)\)
\(\chi_{805}(122,\cdot)\)
\(\chi_{805}(143,\cdot)\)
\(\chi_{805}(152,\cdot)\)
\(\chi_{805}(157,\cdot)\)
\(\chi_{805}(178,\cdot)\)
\(\chi_{805}(222,\cdot)\)
\(\chi_{805}(227,\cdot)\)
\(\chi_{805}(283,\cdot)\)
\(\chi_{805}(297,\cdot)\)
\(\chi_{805}(313,\cdot)\)
\(\chi_{805}(318,\cdot)\)
\(\chi_{805}(327,\cdot)\)
\(\chi_{805}(332,\cdot)\)
\(\chi_{805}(362,\cdot)\)
\(\chi_{805}(383,\cdot)\)
\(\chi_{805}(388,\cdot)\)
\(\chi_{805}(402,\cdot)\)
\(\chi_{805}(458,\cdot)\)
\(\chi_{805}(467,\cdot)\)
\(\chi_{805}(488,\cdot)\)
\(\chi_{805}(493,\cdot)\)
\(\chi_{805}(502,\cdot)\)
\(\chi_{805}(523,\cdot)\)
\(\chi_{805}(563,\cdot)\)
\(\chi_{805}(572,\cdot)\)
\(\chi_{805}(612,\cdot)\)
...
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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 ]
\((162,346,281)\) → \((i,e\left(\frac{1}{6}\right),e\left(\frac{7}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 805 }(17, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{29}{132}\right)\) | \(e\left(\frac{1}{132}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{59}{132}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{29}{33}\right)\) |
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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)
