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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([9,10]))
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pari:[g,chi] = znchar(Mod(2,475))
| 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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{475}(2,\cdot)\)
\(\chi_{475}(3,\cdot)\)
\(\chi_{475}(13,\cdot)\)
\(\chi_{475}(22,\cdot)\)
\(\chi_{475}(33,\cdot)\)
\(\chi_{475}(48,\cdot)\)
\(\chi_{475}(52,\cdot)\)
\(\chi_{475}(53,\cdot)\)
\(\chi_{475}(67,\cdot)\)
\(\chi_{475}(72,\cdot)\)
\(\chi_{475}(78,\cdot)\)
\(\chi_{475}(97,\cdot)\)
\(\chi_{475}(98,\cdot)\)
\(\chi_{475}(108,\cdot)\)
\(\chi_{475}(117,\cdot)\)
\(\chi_{475}(127,\cdot)\)
\(\chi_{475}(128,\cdot)\)
\(\chi_{475}(147,\cdot)\)
\(\chi_{475}(148,\cdot)\)
\(\chi_{475}(162,\cdot)\)
\(\chi_{475}(167,\cdot)\)
\(\chi_{475}(173,\cdot)\)
\(\chi_{475}(192,\cdot)\)
\(\chi_{475}(203,\cdot)\)
\(\chi_{475}(212,\cdot)\)
\(\chi_{475}(222,\cdot)\)
\(\chi_{475}(223,\cdot)\)
\(\chi_{475}(238,\cdot)\)
\(\chi_{475}(242,\cdot)\)
\(\chi_{475}(262,\cdot)\)
...
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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 ]
\((77,401)\) → \((e\left(\frac{1}{20}\right),e\left(\frac{1}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 475 }(2, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{180}\right)\) | \(e\left(\frac{13}{180}\right)\) | \(e\left(\frac{19}{90}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{13}{90}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{41}{180}\right)\) |
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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)
