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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3168, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,3,16,12]))
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pari:[g,chi] = znchar(Mod(1627,3168))
| Modulus: | \(3168\) | |
| Conductor: | \(3168\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| 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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\(\chi_{3168}(43,\cdot)\)
\(\chi_{3168}(571,\cdot)\)
\(\chi_{3168}(835,\cdot)\)
\(\chi_{3168}(1363,\cdot)\)
\(\chi_{3168}(1627,\cdot)\)
\(\chi_{3168}(2155,\cdot)\)
\(\chi_{3168}(2419,\cdot)\)
\(\chi_{3168}(2947,\cdot)\)
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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 ]
\((991,1189,353,1729)\) → \((-1,e\left(\frac{1}{8}\right),e\left(\frac{2}{3}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 3168 }(1627, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{3}{8}\right)\) |
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sage:chi.jacobi_sum(n)
