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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([27,25]))
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pari:[g,chi] = znchar(Mod(89,475))
| 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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{475}(14,\cdot)\)
\(\chi_{475}(29,\cdot)\)
\(\chi_{475}(34,\cdot)\)
\(\chi_{475}(59,\cdot)\)
\(\chi_{475}(79,\cdot)\)
\(\chi_{475}(89,\cdot)\)
\(\chi_{475}(109,\cdot)\)
\(\chi_{475}(129,\cdot)\)
\(\chi_{475}(154,\cdot)\)
\(\chi_{475}(184,\cdot)\)
\(\chi_{475}(204,\cdot)\)
\(\chi_{475}(219,\cdot)\)
\(\chi_{475}(269,\cdot)\)
\(\chi_{475}(279,\cdot)\)
\(\chi_{475}(314,\cdot)\)
\(\chi_{475}(319,\cdot)\)
\(\chi_{475}(344,\cdot)\)
\(\chi_{475}(364,\cdot)\)
\(\chi_{475}(394,\cdot)\)
\(\chi_{475}(409,\cdot)\)
\(\chi_{475}(414,\cdot)\)
\(\chi_{475}(439,\cdot)\)
\(\chi_{475}(459,\cdot)\)
\(\chi_{475}(469,\cdot)\)
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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 ]
\((77,401)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{5}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 475 }(89, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{4}{45}\right)\) |
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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)
