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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(763, base_ring=CyclotomicField(108))
M = H._module
chi = DirichletCharacter(H, M([90,79]))
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pari:[g,chi] = znchar(Mod(208,763))
| Modulus: | \(763\) | |
| Conductor: | \(763\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(108\) |
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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)
|
\(\chi_{763}(40,\cdot)\)
\(\chi_{763}(59,\cdot)\)
\(\chi_{763}(96,\cdot)\)
\(\chi_{763}(103,\cdot)\)
\(\chi_{763}(166,\cdot)\)
\(\chi_{763}(171,\cdot)\)
\(\chi_{763}(194,\cdot)\)
\(\chi_{763}(208,\cdot)\)
\(\chi_{763}(248,\cdot)\)
\(\chi_{763}(255,\cdot)\)
\(\chi_{763}(262,\cdot)\)
\(\chi_{763}(271,\cdot)\)
\(\chi_{763}(283,\cdot)\)
\(\chi_{763}(290,\cdot)\)
\(\chi_{763}(297,\cdot)\)
\(\chi_{763}(313,\cdot)\)
\(\chi_{763}(341,\cdot)\)
\(\chi_{763}(374,\cdot)\)
\(\chi_{763}(383,\cdot)\)
\(\chi_{763}(446,\cdot)\)
\(\chi_{763}(460,\cdot)\)
\(\chi_{763}(486,\cdot)\)
\(\chi_{763}(488,\cdot)\)
\(\chi_{763}(551,\cdot)\)
\(\chi_{763}(556,\cdot)\)
\(\chi_{763}(558,\cdot)\)
\(\chi_{763}(563,\cdot)\)
\(\chi_{763}(584,\cdot)\)
\(\chi_{763}(612,\cdot)\)
\(\chi_{763}(614,\cdot)\)
...
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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 ]
\((437,442)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{79}{108}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 763 }(208, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{25}{108}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{13}{108}\right)\) | \(e\left(\frac{5}{108}\right)\) | \(e\left(\frac{16}{27}\right)\) |
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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)
