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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4176, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([6,3,2,6]))
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pari:[g,chi] = znchar(Mod(1739,4176))
| Modulus: | \(4176\) | |
| Conductor: | \(4176\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(12\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
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| 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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\(\chi_{4176}(347,\cdot)\)
\(\chi_{4176}(1739,\cdot)\)
\(\chi_{4176}(2435,\cdot)\)
\(\chi_{4176}(3827,\cdot)\)
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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 ]
\((1567,1045,929,4033)\) → \((-1,i,e\left(\frac{1}{6}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(31\) | \(35\) |
| \( \chi_{ 4176 }(1739, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(1\) | \(-i\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-i\) |
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sage:chi.jacobi_sum(n)
