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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(475, base_ring=CyclotomicField(90))
M = H._module
chi = DirichletCharacter(H, M([63,50]))
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pari:[g,chi] = znchar(Mod(359,475))
| Modulus: | \(475\) | |
| Conductor: | \(475\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(90\) |
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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
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| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{475}(4,\cdot)\)
\(\chi_{475}(9,\cdot)\)
\(\chi_{475}(44,\cdot)\)
\(\chi_{475}(54,\cdot)\)
\(\chi_{475}(104,\cdot)\)
\(\chi_{475}(119,\cdot)\)
\(\chi_{475}(139,\cdot)\)
\(\chi_{475}(169,\cdot)\)
\(\chi_{475}(194,\cdot)\)
\(\chi_{475}(214,\cdot)\)
\(\chi_{475}(234,\cdot)\)
\(\chi_{475}(244,\cdot)\)
\(\chi_{475}(264,\cdot)\)
\(\chi_{475}(289,\cdot)\)
\(\chi_{475}(294,\cdot)\)
\(\chi_{475}(309,\cdot)\)
\(\chi_{475}(329,\cdot)\)
\(\chi_{475}(339,\cdot)\)
\(\chi_{475}(359,\cdot)\)
\(\chi_{475}(384,\cdot)\)
\(\chi_{475}(389,\cdot)\)
\(\chi_{475}(404,\cdot)\)
\(\chi_{475}(434,\cdot)\)
\(\chi_{475}(454,\cdot)\)
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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 ]
\((77,401)\) → \((e\left(\frac{7}{10}\right),e\left(\frac{5}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 475 }(359, a) \) |
\(1\) | \(1\) | \(e\left(\frac{23}{90}\right)\) | \(e\left(\frac{11}{90}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{7}{90}\right)\) |
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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)
