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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(269, base_ring=CyclotomicField(134))
M = H._module
chi = DirichletCharacter(H, M([20]))
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pari:[g,chi] = znchar(Mod(196,269))
| Modulus: | \(269\) | |
| Conductor: | \(269\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(67\) |
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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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\(\chi_{269}(5,\cdot)\)
\(\chi_{269}(14,\cdot)\)
\(\chi_{269}(16,\cdot)\)
\(\chi_{269}(21,\cdot)\)
\(\chi_{269}(23,\cdot)\)
\(\chi_{269}(24,\cdot)\)
\(\chi_{269}(25,\cdot)\)
\(\chi_{269}(36,\cdot)\)
\(\chi_{269}(37,\cdot)\)
\(\chi_{269}(38,\cdot)\)
\(\chi_{269}(41,\cdot)\)
\(\chi_{269}(44,\cdot)\)
\(\chi_{269}(47,\cdot)\)
\(\chi_{269}(52,\cdot)\)
\(\chi_{269}(53,\cdot)\)
\(\chi_{269}(54,\cdot)\)
\(\chi_{269}(57,\cdot)\)
\(\chi_{269}(58,\cdot)\)
\(\chi_{269}(61,\cdot)\)
\(\chi_{269}(62,\cdot)\)
\(\chi_{269}(66,\cdot)\)
\(\chi_{269}(67,\cdot)\)
\(\chi_{269}(70,\cdot)\)
\(\chi_{269}(78,\cdot)\)
\(\chi_{269}(80,\cdot)\)
\(\chi_{269}(81,\cdot)\)
\(\chi_{269}(87,\cdot)\)
\(\chi_{269}(93,\cdot)\)
\(\chi_{269}(99,\cdot)\)
\(\chi_{269}(105,\cdot)\)
...
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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 ]
\(2\) → \(e\left(\frac{10}{67}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 269 }(196, a) \) |
\(1\) | \(1\) | \(e\left(\frac{10}{67}\right)\) | \(e\left(\frac{18}{67}\right)\) | \(e\left(\frac{20}{67}\right)\) | \(e\left(\frac{3}{67}\right)\) | \(e\left(\frac{28}{67}\right)\) | \(e\left(\frac{56}{67}\right)\) | \(e\left(\frac{30}{67}\right)\) | \(e\left(\frac{36}{67}\right)\) | \(e\left(\frac{13}{67}\right)\) | \(e\left(\frac{22}{67}\right)\) |
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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)
