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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1083, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,13]))
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pari:[g,chi] = znchar(Mod(170,1083))
| Modulus: | \(1083\) | |
| Conductor: | \(1083\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(38\) |
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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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\(\chi_{1083}(56,\cdot)\)
\(\chi_{1083}(113,\cdot)\)
\(\chi_{1083}(170,\cdot)\)
\(\chi_{1083}(227,\cdot)\)
\(\chi_{1083}(284,\cdot)\)
\(\chi_{1083}(341,\cdot)\)
\(\chi_{1083}(398,\cdot)\)
\(\chi_{1083}(455,\cdot)\)
\(\chi_{1083}(512,\cdot)\)
\(\chi_{1083}(569,\cdot)\)
\(\chi_{1083}(626,\cdot)\)
\(\chi_{1083}(683,\cdot)\)
\(\chi_{1083}(740,\cdot)\)
\(\chi_{1083}(797,\cdot)\)
\(\chi_{1083}(854,\cdot)\)
\(\chi_{1083}(911,\cdot)\)
\(\chi_{1083}(968,\cdot)\)
\(\chi_{1083}(1025,\cdot)\)
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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 ]
\((362,724)\) → \((-1,e\left(\frac{13}{38}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 1083 }(170, a) \) |
\(1\) | \(1\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) |
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sage:chi.jacobi_sum(n)
