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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(459, base_ring=CyclotomicField(144))
M = H._module
chi = DirichletCharacter(H, M([16,81]))
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pari:[g,chi] = znchar(Mod(31,459))
| Modulus: | \(459\) | |
| Conductor: | \(459\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(144\) |
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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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\(\chi_{459}(7,\cdot)\)
\(\chi_{459}(22,\cdot)\)
\(\chi_{459}(31,\cdot)\)
\(\chi_{459}(40,\cdot)\)
\(\chi_{459}(58,\cdot)\)
\(\chi_{459}(61,\cdot)\)
\(\chi_{459}(79,\cdot)\)
\(\chi_{459}(88,\cdot)\)
\(\chi_{459}(97,\cdot)\)
\(\chi_{459}(112,\cdot)\)
\(\chi_{459}(124,\cdot)\)
\(\chi_{459}(130,\cdot)\)
\(\chi_{459}(133,\cdot)\)
\(\chi_{459}(139,\cdot)\)
\(\chi_{459}(142,\cdot)\)
\(\chi_{459}(148,\cdot)\)
\(\chi_{459}(160,\cdot)\)
\(\chi_{459}(175,\cdot)\)
\(\chi_{459}(184,\cdot)\)
\(\chi_{459}(193,\cdot)\)
\(\chi_{459}(211,\cdot)\)
\(\chi_{459}(214,\cdot)\)
\(\chi_{459}(232,\cdot)\)
\(\chi_{459}(241,\cdot)\)
\(\chi_{459}(250,\cdot)\)
\(\chi_{459}(265,\cdot)\)
\(\chi_{459}(277,\cdot)\)
\(\chi_{459}(283,\cdot)\)
\(\chi_{459}(286,\cdot)\)
\(\chi_{459}(292,\cdot)\)
...
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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 ]
\((137,190)\) → \((e\left(\frac{1}{9}\right),e\left(\frac{9}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 459 }(31, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{71}{72}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{53}{144}\right)\) | \(e\left(\frac{139}{144}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{55}{144}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{137}{144}\right)\) | \(e\left(\frac{17}{18}\right)\) |
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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)
