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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(512, base_ring=CyclotomicField(128))
M = H._module
chi = DirichletCharacter(H, M([0,5]))
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pari:[g,chi] = znchar(Mod(53,512))
| Modulus: | \(512\) | |
| Conductor: | \(512\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(128\) |
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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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\(\chi_{512}(5,\cdot)\)
\(\chi_{512}(13,\cdot)\)
\(\chi_{512}(21,\cdot)\)
\(\chi_{512}(29,\cdot)\)
\(\chi_{512}(37,\cdot)\)
\(\chi_{512}(45,\cdot)\)
\(\chi_{512}(53,\cdot)\)
\(\chi_{512}(61,\cdot)\)
\(\chi_{512}(69,\cdot)\)
\(\chi_{512}(77,\cdot)\)
\(\chi_{512}(85,\cdot)\)
\(\chi_{512}(93,\cdot)\)
\(\chi_{512}(101,\cdot)\)
\(\chi_{512}(109,\cdot)\)
\(\chi_{512}(117,\cdot)\)
\(\chi_{512}(125,\cdot)\)
\(\chi_{512}(133,\cdot)\)
\(\chi_{512}(141,\cdot)\)
\(\chi_{512}(149,\cdot)\)
\(\chi_{512}(157,\cdot)\)
\(\chi_{512}(165,\cdot)\)
\(\chi_{512}(173,\cdot)\)
\(\chi_{512}(181,\cdot)\)
\(\chi_{512}(189,\cdot)\)
\(\chi_{512}(197,\cdot)\)
\(\chi_{512}(205,\cdot)\)
\(\chi_{512}(213,\cdot)\)
\(\chi_{512}(221,\cdot)\)
\(\chi_{512}(229,\cdot)\)
\(\chi_{512}(237,\cdot)\)
...
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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 ]
\((511,5)\) → \((1,e\left(\frac{5}{128}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 512 }(53, a) \) |
\(1\) | \(1\) | \(e\left(\frac{47}{128}\right)\) | \(e\left(\frac{5}{128}\right)\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{41}{128}\right)\) | \(e\left(\frac{43}{128}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{115}{128}\right)\) | \(e\left(\frac{33}{128}\right)\) |
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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)
