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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(261, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([28,57]))
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pari:[g,chi] = znchar(Mod(229,261))
| Modulus: | \(261\) | |
| Conductor: | \(261\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{261}(31,\cdot)\)
\(\chi_{261}(40,\cdot)\)
\(\chi_{261}(43,\cdot)\)
\(\chi_{261}(61,\cdot)\)
\(\chi_{261}(76,\cdot)\)
\(\chi_{261}(79,\cdot)\)
\(\chi_{261}(85,\cdot)\)
\(\chi_{261}(97,\cdot)\)
\(\chi_{261}(106,\cdot)\)
\(\chi_{261}(124,\cdot)\)
\(\chi_{261}(130,\cdot)\)
\(\chi_{261}(142,\cdot)\)
\(\chi_{261}(148,\cdot)\)
\(\chi_{261}(160,\cdot)\)
\(\chi_{261}(166,\cdot)\)
\(\chi_{261}(184,\cdot)\)
\(\chi_{261}(193,\cdot)\)
\(\chi_{261}(205,\cdot)\)
\(\chi_{261}(211,\cdot)\)
\(\chi_{261}(214,\cdot)\)
\(\chi_{261}(229,\cdot)\)
\(\chi_{261}(247,\cdot)\)
\(\chi_{261}(250,\cdot)\)
\(\chi_{261}(259,\cdot)\)
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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 ]
\((146,118)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{19}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 261 }(229, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{1}{21}\right)\) |
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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)
