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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(784, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,7,10]))
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pari:[g,chi] = znchar(Mod(139,784))
| Modulus: | \(784\) | |
| Conductor: | \(784\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(28\) |
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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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\(\chi_{784}(27,\cdot)\)
\(\chi_{784}(83,\cdot)\)
\(\chi_{784}(139,\cdot)\)
\(\chi_{784}(251,\cdot)\)
\(\chi_{784}(307,\cdot)\)
\(\chi_{784}(363,\cdot)\)
\(\chi_{784}(419,\cdot)\)
\(\chi_{784}(475,\cdot)\)
\(\chi_{784}(531,\cdot)\)
\(\chi_{784}(643,\cdot)\)
\(\chi_{784}(699,\cdot)\)
\(\chi_{784}(755,\cdot)\)
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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 ]
\((687,197,689)\) → \((-1,i,e\left(\frac{5}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 784 }(139, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(-i\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{3}{14}\right)\) |
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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)
