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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1288, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,11,11,9]))
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pari:[g,chi] = znchar(Mod(517,1288))
| Modulus: | \(1288\) | |
| Conductor: | \(1288\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(22\) |
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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
|
| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{1288}(125,\cdot)\)
\(\chi_{1288}(181,\cdot)\)
\(\chi_{1288}(237,\cdot)\)
\(\chi_{1288}(293,\cdot)\)
\(\chi_{1288}(405,\cdot)\)
\(\chi_{1288}(517,\cdot)\)
\(\chi_{1288}(573,\cdot)\)
\(\chi_{1288}(741,\cdot)\)
\(\chi_{1288}(797,\cdot)\)
\(\chi_{1288}(1077,\cdot)\)
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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 ]
\((967,645,185,281)\) → \((1,-1,-1,e\left(\frac{9}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(25\) | \(27\) |
| \( \chi_{ 1288 }(517, a) \) |
\(1\) | \(1\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) |
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sage:chi.jacobi_sum(n)
