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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(784, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([0,21,20]))
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pari:[g,chi] = znchar(Mod(53,784))
| Modulus: | \(784\) | |
| Conductor: | \(784\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(84\) |
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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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\(\chi_{784}(37,\cdot)\)
\(\chi_{784}(53,\cdot)\)
\(\chi_{784}(93,\cdot)\)
\(\chi_{784}(109,\cdot)\)
\(\chi_{784}(149,\cdot)\)
\(\chi_{784}(205,\cdot)\)
\(\chi_{784}(221,\cdot)\)
\(\chi_{784}(261,\cdot)\)
\(\chi_{784}(277,\cdot)\)
\(\chi_{784}(317,\cdot)\)
\(\chi_{784}(333,\cdot)\)
\(\chi_{784}(389,\cdot)\)
\(\chi_{784}(429,\cdot)\)
\(\chi_{784}(445,\cdot)\)
\(\chi_{784}(485,\cdot)\)
\(\chi_{784}(501,\cdot)\)
\(\chi_{784}(541,\cdot)\)
\(\chi_{784}(597,\cdot)\)
\(\chi_{784}(613,\cdot)\)
\(\chi_{784}(653,\cdot)\)
\(\chi_{784}(669,\cdot)\)
\(\chi_{784}(709,\cdot)\)
\(\chi_{784}(725,\cdot)\)
\(\chi_{784}(781,\cdot)\)
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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 ]
\((687,197,689)\) → \((1,i,e\left(\frac{5}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 784 }(53, a) \) |
\(1\) | \(1\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{13}{42}\right)\) |
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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)
