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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([9]))
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pari:[g,chi] = znchar(Mod(153,343))
| Modulus: | \(343\) | |
| Conductor: | \(343\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(98\) |
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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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\(\chi_{343}(6,\cdot)\)
\(\chi_{343}(13,\cdot)\)
\(\chi_{343}(20,\cdot)\)
\(\chi_{343}(27,\cdot)\)
\(\chi_{343}(34,\cdot)\)
\(\chi_{343}(41,\cdot)\)
\(\chi_{343}(55,\cdot)\)
\(\chi_{343}(62,\cdot)\)
\(\chi_{343}(69,\cdot)\)
\(\chi_{343}(76,\cdot)\)
\(\chi_{343}(83,\cdot)\)
\(\chi_{343}(90,\cdot)\)
\(\chi_{343}(104,\cdot)\)
\(\chi_{343}(111,\cdot)\)
\(\chi_{343}(118,\cdot)\)
\(\chi_{343}(125,\cdot)\)
\(\chi_{343}(132,\cdot)\)
\(\chi_{343}(139,\cdot)\)
\(\chi_{343}(153,\cdot)\)
\(\chi_{343}(160,\cdot)\)
\(\chi_{343}(167,\cdot)\)
\(\chi_{343}(174,\cdot)\)
\(\chi_{343}(181,\cdot)\)
\(\chi_{343}(188,\cdot)\)
\(\chi_{343}(202,\cdot)\)
\(\chi_{343}(209,\cdot)\)
\(\chi_{343}(216,\cdot)\)
\(\chi_{343}(223,\cdot)\)
\(\chi_{343}(230,\cdot)\)
\(\chi_{343}(237,\cdot)\)
...
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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 ]
\(3\) → \(e\left(\frac{9}{98}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 343 }(153, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{40}{49}\right)\) | \(e\left(\frac{9}{98}\right)\) | \(e\left(\frac{31}{49}\right)\) | \(e\left(\frac{65}{98}\right)\) | \(e\left(\frac{89}{98}\right)\) | \(e\left(\frac{22}{49}\right)\) | \(e\left(\frac{9}{49}\right)\) | \(e\left(\frac{47}{98}\right)\) | \(e\left(\frac{5}{49}\right)\) | \(e\left(\frac{71}{98}\right)\) |
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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)
