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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1120, base_ring=CyclotomicField(8))
M = H._module
chi = DirichletCharacter(H, M([0,3,2,4]))
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pari:[g,chi] = znchar(Mod(797,1120))
| Modulus: | \(1120\) | |
| Conductor: | \(1120\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(8\) |
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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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\(\chi_{1120}(237,\cdot)\)
\(\chi_{1120}(293,\cdot)\)
\(\chi_{1120}(797,\cdot)\)
\(\chi_{1120}(853,\cdot)\)
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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 ]
\((351,421,897,801)\) → \((1,e\left(\frac{3}{8}\right),i,-1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 1120 }(797, a) \) |
\(1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(-i\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(i\) | \(e\left(\frac{5}{8}\right)\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(-1\) |
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sage:chi.jacobi_sum(n)
