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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(343, base_ring=CyclotomicField(294))
M = H._module
chi = DirichletCharacter(H, M([95]))
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pari:[g,chi] = znchar(Mod(12,343))
| Modulus: | \(343\) | |
| Conductor: | \(343\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(294\) |
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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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\(\chi_{343}(3,\cdot)\)
\(\chi_{343}(5,\cdot)\)
\(\chi_{343}(10,\cdot)\)
\(\chi_{343}(12,\cdot)\)
\(\chi_{343}(17,\cdot)\)
\(\chi_{343}(24,\cdot)\)
\(\chi_{343}(26,\cdot)\)
\(\chi_{343}(33,\cdot)\)
\(\chi_{343}(38,\cdot)\)
\(\chi_{343}(40,\cdot)\)
\(\chi_{343}(45,\cdot)\)
\(\chi_{343}(47,\cdot)\)
\(\chi_{343}(52,\cdot)\)
\(\chi_{343}(54,\cdot)\)
\(\chi_{343}(59,\cdot)\)
\(\chi_{343}(61,\cdot)\)
\(\chi_{343}(66,\cdot)\)
\(\chi_{343}(73,\cdot)\)
\(\chi_{343}(75,\cdot)\)
\(\chi_{343}(82,\cdot)\)
\(\chi_{343}(87,\cdot)\)
\(\chi_{343}(89,\cdot)\)
\(\chi_{343}(94,\cdot)\)
\(\chi_{343}(96,\cdot)\)
\(\chi_{343}(101,\cdot)\)
\(\chi_{343}(103,\cdot)\)
\(\chi_{343}(108,\cdot)\)
\(\chi_{343}(110,\cdot)\)
\(\chi_{343}(115,\cdot)\)
\(\chi_{343}(122,\cdot)\)
...
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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 ]
\(3\) → \(e\left(\frac{95}{294}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 343 }(12, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{101}{147}\right)\) | \(e\left(\frac{95}{294}\right)\) | \(e\left(\frac{55}{147}\right)\) | \(e\left(\frac{109}{294}\right)\) | \(e\left(\frac{1}{98}\right)\) | \(e\left(\frac{3}{49}\right)\) | \(e\left(\frac{95}{147}\right)\) | \(e\left(\frac{17}{294}\right)\) | \(e\left(\frac{31}{147}\right)\) | \(e\left(\frac{205}{294}\right)\) |
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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)
