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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8568, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,12,4,8,3]))
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pari:[g,chi] = znchar(Mod(5483,8568))
| Modulus: | \(8568\) | |
| Conductor: | \(8568\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(24\) |
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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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{8568}(4979,\cdot)\)
\(\chi_{8568}(5387,\cdot)\)
\(\chi_{8568}(5483,\cdot)\)
\(\chi_{8568}(5891,\cdot)\)
\(\chi_{8568}(6995,\cdot)\)
\(\chi_{8568}(7403,\cdot)\)
\(\chi_{8568}(7499,\cdot)\)
\(\chi_{8568}(7907,\cdot)\)
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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 ]
\((2143,4285,3809,6121,5545)\) → \((-1,-1,e\left(\frac{1}{6}\right),e\left(\frac{1}{3}\right),e\left(\frac{1}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 8568 }(5483, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(i\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{5}{24}\right)\) |
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sage:chi.jacobi_sum(n)
