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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(833, base_ring=CyclotomicField(168))
M = H._module
chi = DirichletCharacter(H, M([16,63]))
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pari:[g,chi] = znchar(Mod(32,833))
| Modulus: | \(833\) | |
| Conductor: | \(833\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(168\) |
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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)
|
\(\chi_{833}(2,\cdot)\)
\(\chi_{833}(9,\cdot)\)
\(\chi_{833}(25,\cdot)\)
\(\chi_{833}(32,\cdot)\)
\(\chi_{833}(53,\cdot)\)
\(\chi_{833}(60,\cdot)\)
\(\chi_{833}(93,\cdot)\)
\(\chi_{833}(100,\cdot)\)
\(\chi_{833}(121,\cdot)\)
\(\chi_{833}(144,\cdot)\)
\(\chi_{833}(151,\cdot)\)
\(\chi_{833}(172,\cdot)\)
\(\chi_{833}(179,\cdot)\)
\(\chi_{833}(212,\cdot)\)
\(\chi_{833}(219,\cdot)\)
\(\chi_{833}(240,\cdot)\)
\(\chi_{833}(247,\cdot)\)
\(\chi_{833}(270,\cdot)\)
\(\chi_{833}(291,\cdot)\)
\(\chi_{833}(298,\cdot)\)
\(\chi_{833}(331,\cdot)\)
\(\chi_{833}(338,\cdot)\)
\(\chi_{833}(359,\cdot)\)
\(\chi_{833}(366,\cdot)\)
\(\chi_{833}(382,\cdot)\)
\(\chi_{833}(389,\cdot)\)
\(\chi_{833}(417,\cdot)\)
\(\chi_{833}(450,\cdot)\)
\(\chi_{833}(457,\cdot)\)
\(\chi_{833}(478,\cdot)\)
...
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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 ]
\((52,785)\) → \((e\left(\frac{2}{21}\right),e\left(\frac{3}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 833 }(32, a) \) |
\(1\) | \(1\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{79}{168}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{107}{168}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{61}{168}\right)\) | \(e\left(\frac{73}{168}\right)\) | \(e\left(\frac{155}{168}\right)\) |
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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)
