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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(967, base_ring=CyclotomicField(966))
M = H._module
chi = DirichletCharacter(H, M([113]))
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pari:[g,chi] = znchar(Mod(171,967))
| Modulus: | \(967\) | |
| Conductor: | \(967\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(966\) |
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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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\(\chi_{967}(5,\cdot)\)
\(\chi_{967}(6,\cdot)\)
\(\chi_{967}(7,\cdot)\)
\(\chi_{967}(12,\cdot)\)
\(\chi_{967}(13,\cdot)\)
\(\chi_{967}(19,\cdot)\)
\(\chi_{967}(28,\cdot)\)
\(\chi_{967}(37,\cdot)\)
\(\chi_{967}(38,\cdot)\)
\(\chi_{967}(40,\cdot)\)
\(\chi_{967}(43,\cdot)\)
\(\chi_{967}(45,\cdot)\)
\(\chi_{967}(46,\cdot)\)
\(\chi_{967}(47,\cdot)\)
\(\chi_{967}(48,\cdot)\)
\(\chi_{967}(56,\cdot)\)
\(\chi_{967}(58,\cdot)\)
\(\chi_{967}(63,\cdot)\)
\(\chi_{967}(66,\cdot)\)
\(\chi_{967}(75,\cdot)\)
\(\chi_{967}(77,\cdot)\)
\(\chi_{967}(79,\cdot)\)
\(\chi_{967}(82,\cdot)\)
\(\chi_{967}(85,\cdot)\)
\(\chi_{967}(86,\cdot)\)
\(\chi_{967}(89,\cdot)\)
\(\chi_{967}(102,\cdot)\)
\(\chi_{967}(104,\cdot)\)
\(\chi_{967}(105,\cdot)\)
\(\chi_{967}(107,\cdot)\)
...
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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 ]
\(5\) → \(e\left(\frac{113}{966}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 967 }(171, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{362}{483}\right)\) | \(e\left(\frac{233}{322}\right)\) | \(e\left(\frac{241}{483}\right)\) | \(e\left(\frac{113}{966}\right)\) | \(e\left(\frac{457}{966}\right)\) | \(e\left(\frac{809}{966}\right)\) | \(e\left(\frac{40}{161}\right)\) | \(e\left(\frac{72}{161}\right)\) | \(e\left(\frac{279}{322}\right)\) | \(e\left(\frac{141}{161}\right)\) |
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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)
