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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(34272, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,21,8,4,9]))
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pari:[g,chi] = znchar(Mod(2803,34272))
| Modulus: | \(34272\) | |
| Conductor: | \(34272\) |
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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)
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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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\(\chi_{34272}(2803,\cdot)\)
\(\chi_{34272}(3307,\cdot)\)
\(\chi_{34272}(5731,\cdot)\)
\(\chi_{34272}(13891,\cdot)\)
\(\chi_{34272}(22363,\cdot)\)
\(\chi_{34272}(28915,\cdot)\)
\(\chi_{34272}(29419,\cdot)\)
\(\chi_{34272}(30523,\cdot)\)
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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 ]
\((2143,29989,3809,14689,14113)\) → \((-1,e\left(\frac{7}{8}\right),e\left(\frac{1}{3}\right),e\left(\frac{1}{6}\right),e\left(\frac{3}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 34272 }(2803, a) \) |
\(1\) | \(1\) | \(i\) | \(-1\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(-1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{13}{24}\right)\) |
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sage:chi.jacobi_sum(n)
