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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(645, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([42,63,8]))
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pari:[g,chi] = znchar(Mod(38,645))
| Modulus: | \(645\) | |
| Conductor: | \(645\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(84\) |
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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
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| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{645}(17,\cdot)\)
\(\chi_{645}(23,\cdot)\)
\(\chi_{645}(38,\cdot)\)
\(\chi_{645}(53,\cdot)\)
\(\chi_{645}(68,\cdot)\)
\(\chi_{645}(83,\cdot)\)
\(\chi_{645}(143,\cdot)\)
\(\chi_{645}(152,\cdot)\)
\(\chi_{645}(167,\cdot)\)
\(\chi_{645}(182,\cdot)\)
\(\chi_{645}(197,\cdot)\)
\(\chi_{645}(203,\cdot)\)
\(\chi_{645}(212,\cdot)\)
\(\chi_{645}(272,\cdot)\)
\(\chi_{645}(332,\cdot)\)
\(\chi_{645}(353,\cdot)\)
\(\chi_{645}(368,\cdot)\)
\(\chi_{645}(443,\cdot)\)
\(\chi_{645}(482,\cdot)\)
\(\chi_{645}(488,\cdot)\)
\(\chi_{645}(497,\cdot)\)
\(\chi_{645}(533,\cdot)\)
\(\chi_{645}(572,\cdot)\)
\(\chi_{645}(617,\cdot)\)
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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 ]
\((431,517,46)\) → \((-1,-i,e\left(\frac{2}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 645 }(38, a) \) |
\(1\) | \(1\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{13}{42}\right)\) |
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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)
