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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(664, base_ring=CyclotomicField(82))
M = H._module
chi = DirichletCharacter(H, M([41,41,11]))
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pari:[g,chi] = znchar(Mod(139,664))
| Modulus: | \(664\) | |
| Conductor: | \(664\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(82\) |
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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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\(\chi_{664}(19,\cdot)\)
\(\chi_{664}(35,\cdot)\)
\(\chi_{664}(43,\cdot)\)
\(\chi_{664}(67,\cdot)\)
\(\chi_{664}(91,\cdot)\)
\(\chi_{664}(107,\cdot)\)
\(\chi_{664}(115,\cdot)\)
\(\chi_{664}(139,\cdot)\)
\(\chi_{664}(155,\cdot)\)
\(\chi_{664}(163,\cdot)\)
\(\chi_{664}(171,\cdot)\)
\(\chi_{664}(179,\cdot)\)
\(\chi_{664}(211,\cdot)\)
\(\chi_{664}(219,\cdot)\)
\(\chi_{664}(251,\cdot)\)
\(\chi_{664}(267,\cdot)\)
\(\chi_{664}(283,\cdot)\)
\(\chi_{664}(291,\cdot)\)
\(\chi_{664}(299,\cdot)\)
\(\chi_{664}(307,\cdot)\)
\(\chi_{664}(315,\cdot)\)
\(\chi_{664}(323,\cdot)\)
\(\chi_{664}(347,\cdot)\)
\(\chi_{664}(371,\cdot)\)
\(\chi_{664}(379,\cdot)\)
\(\chi_{664}(387,\cdot)\)
\(\chi_{664}(403,\cdot)\)
\(\chi_{664}(411,\cdot)\)
\(\chi_{664}(435,\cdot)\)
\(\chi_{664}(467,\cdot)\)
...
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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 ]
\((167,333,417)\) → \((-1,-1,e\left(\frac{11}{82}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 664 }(139, a) \) |
\(1\) | \(1\) | \(e\left(\frac{27}{41}\right)\) | \(e\left(\frac{5}{41}\right)\) | \(e\left(\frac{47}{82}\right)\) | \(e\left(\frac{13}{41}\right)\) | \(e\left(\frac{9}{41}\right)\) | \(e\left(\frac{34}{41}\right)\) | \(e\left(\frac{32}{41}\right)\) | \(e\left(\frac{21}{41}\right)\) | \(e\left(\frac{25}{82}\right)\) | \(e\left(\frac{19}{82}\right)\) |
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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)
