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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(343, base_ring=CyclotomicField(98))
M = H._module
chi = DirichletCharacter(H, M([92]))
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pari:[g,chi] = znchar(Mod(169,343))
| Modulus: | \(343\) | |
| Conductor: | \(343\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(49\) |
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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)
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\(\chi_{343}(8,\cdot)\)
\(\chi_{343}(15,\cdot)\)
\(\chi_{343}(22,\cdot)\)
\(\chi_{343}(29,\cdot)\)
\(\chi_{343}(36,\cdot)\)
\(\chi_{343}(43,\cdot)\)
\(\chi_{343}(57,\cdot)\)
\(\chi_{343}(64,\cdot)\)
\(\chi_{343}(71,\cdot)\)
\(\chi_{343}(78,\cdot)\)
\(\chi_{343}(85,\cdot)\)
\(\chi_{343}(92,\cdot)\)
\(\chi_{343}(106,\cdot)\)
\(\chi_{343}(113,\cdot)\)
\(\chi_{343}(120,\cdot)\)
\(\chi_{343}(127,\cdot)\)
\(\chi_{343}(134,\cdot)\)
\(\chi_{343}(141,\cdot)\)
\(\chi_{343}(155,\cdot)\)
\(\chi_{343}(162,\cdot)\)
\(\chi_{343}(169,\cdot)\)
\(\chi_{343}(176,\cdot)\)
\(\chi_{343}(183,\cdot)\)
\(\chi_{343}(190,\cdot)\)
\(\chi_{343}(204,\cdot)\)
\(\chi_{343}(211,\cdot)\)
\(\chi_{343}(218,\cdot)\)
\(\chi_{343}(225,\cdot)\)
\(\chi_{343}(232,\cdot)\)
\(\chi_{343}(239,\cdot)\)
...
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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 ]
\(3\) → \(e\left(\frac{46}{49}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 343 }(169, a) \) |
\(1\) | \(1\) | \(e\left(\frac{6}{49}\right)\) | \(e\left(\frac{46}{49}\right)\) | \(e\left(\frac{12}{49}\right)\) | \(e\left(\frac{11}{49}\right)\) | \(e\left(\frac{3}{49}\right)\) | \(e\left(\frac{18}{49}\right)\) | \(e\left(\frac{43}{49}\right)\) | \(e\left(\frac{17}{49}\right)\) | \(e\left(\frac{13}{49}\right)\) | \(e\left(\frac{9}{49}\right)\) |
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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)
