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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(475, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([117,20]))
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pari:[g,chi] = znchar(Mod(42,475))
| Modulus: | \(475\) | |
| Conductor: | \(475\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(180\) |
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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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\(\chi_{475}(17,\cdot)\)
\(\chi_{475}(23,\cdot)\)
\(\chi_{475}(28,\cdot)\)
\(\chi_{475}(42,\cdot)\)
\(\chi_{475}(47,\cdot)\)
\(\chi_{475}(62,\cdot)\)
\(\chi_{475}(63,\cdot)\)
\(\chi_{475}(73,\cdot)\)
\(\chi_{475}(92,\cdot)\)
\(\chi_{475}(112,\cdot)\)
\(\chi_{475}(123,\cdot)\)
\(\chi_{475}(137,\cdot)\)
\(\chi_{475}(138,\cdot)\)
\(\chi_{475}(142,\cdot)\)
\(\chi_{475}(158,\cdot)\)
\(\chi_{475}(177,\cdot)\)
\(\chi_{475}(187,\cdot)\)
\(\chi_{475}(188,\cdot)\)
\(\chi_{475}(213,\cdot)\)
\(\chi_{475}(233,\cdot)\)
\(\chi_{475}(237,\cdot)\)
\(\chi_{475}(252,\cdot)\)
\(\chi_{475}(253,\cdot)\)
\(\chi_{475}(263,\cdot)\)
\(\chi_{475}(272,\cdot)\)
\(\chi_{475}(283,\cdot)\)
\(\chi_{475}(302,\cdot)\)
\(\chi_{475}(308,\cdot)\)
\(\chi_{475}(313,\cdot)\)
\(\chi_{475}(327,\cdot)\)
...
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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 ]
\((77,401)\) → \((e\left(\frac{13}{20}\right),e\left(\frac{1}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 475 }(42, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{137}{180}\right)\) | \(e\left(\frac{179}{180}\right)\) | \(e\left(\frac{47}{90}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{89}{90}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{163}{180}\right)\) |
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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)
