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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(847, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([110,3]))
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pari:[g,chi] = znchar(Mod(2,847))
| Modulus: | \(847\) | |
| Conductor: | \(847\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(330\) |
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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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\(\chi_{847}(2,\cdot)\)
\(\chi_{847}(18,\cdot)\)
\(\chi_{847}(30,\cdot)\)
\(\chi_{847}(39,\cdot)\)
\(\chi_{847}(46,\cdot)\)
\(\chi_{847}(51,\cdot)\)
\(\chi_{847}(72,\cdot)\)
\(\chi_{847}(74,\cdot)\)
\(\chi_{847}(79,\cdot)\)
\(\chi_{847}(95,\cdot)\)
\(\chi_{847}(107,\cdot)\)
\(\chi_{847}(116,\cdot)\)
\(\chi_{847}(123,\cdot)\)
\(\chi_{847}(128,\cdot)\)
\(\chi_{847}(149,\cdot)\)
\(\chi_{847}(151,\cdot)\)
\(\chi_{847}(156,\cdot)\)
\(\chi_{847}(172,\cdot)\)
\(\chi_{847}(184,\cdot)\)
\(\chi_{847}(193,\cdot)\)
\(\chi_{847}(200,\cdot)\)
\(\chi_{847}(205,\cdot)\)
\(\chi_{847}(226,\cdot)\)
\(\chi_{847}(228,\cdot)\)
\(\chi_{847}(249,\cdot)\)
\(\chi_{847}(261,\cdot)\)
\(\chi_{847}(270,\cdot)\)
\(\chi_{847}(277,\cdot)\)
\(\chi_{847}(303,\cdot)\)
\(\chi_{847}(305,\cdot)\)
...
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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 ]
\((122,365)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{1}{110}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 847 }(2, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{223}{330}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{58}{165}\right)\) | \(e\left(\frac{56}{165}\right)\) | \(e\left(\frac{89}{110}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{101}{110}\right)\) |
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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)
