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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(687, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,5]))
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pari:[g,chi] = znchar(Mod(644,687))
| Modulus: | \(687\) | |
| Conductor: | \(687\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| 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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{687}(11,\cdot)\)
\(\chi_{687}(26,\cdot)\)
\(\chi_{687}(68,\cdot)\)
\(\chi_{687}(125,\cdot)\)
\(\chi_{687}(176,\cdot)\)
\(\chi_{687}(185,\cdot)\)
\(\chi_{687}(212,\cdot)\)
\(\chi_{687}(233,\cdot)\)
\(\chi_{687}(293,\cdot)\)
\(\chi_{687}(398,\cdot)\)
\(\chi_{687}(401,\cdot)\)
\(\chi_{687}(416,\cdot)\)
\(\chi_{687}(431,\cdot)\)
\(\chi_{687}(473,\cdot)\)
\(\chi_{687}(566,\cdot)\)
\(\chi_{687}(626,\cdot)\)
\(\chi_{687}(644,\cdot)\)
\(\chi_{687}(671,\cdot)\)
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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 ]
\((230,235)\) → \((-1,e\left(\frac{5}{38}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 687 }(644, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{1}{19}\right)\) |
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sage:chi.jacobi_sum(n)
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sage:chi.gauss_sum(a)
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pari:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
