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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(833, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([20,63]))
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pari:[g,chi] = znchar(Mod(4,833))
| Modulus: | \(833\) | |
| Conductor: | \(833\) |
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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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\(\chi_{833}(4,\cdot)\)
\(\chi_{833}(72,\cdot)\)
\(\chi_{833}(81,\cdot)\)
\(\chi_{833}(123,\cdot)\)
\(\chi_{833}(149,\cdot)\)
\(\chi_{833}(191,\cdot)\)
\(\chi_{833}(200,\cdot)\)
\(\chi_{833}(242,\cdot)\)
\(\chi_{833}(268,\cdot)\)
\(\chi_{833}(310,\cdot)\)
\(\chi_{833}(319,\cdot)\)
\(\chi_{833}(387,\cdot)\)
\(\chi_{833}(429,\cdot)\)
\(\chi_{833}(438,\cdot)\)
\(\chi_{833}(480,\cdot)\)
\(\chi_{833}(506,\cdot)\)
\(\chi_{833}(548,\cdot)\)
\(\chi_{833}(599,\cdot)\)
\(\chi_{833}(625,\cdot)\)
\(\chi_{833}(676,\cdot)\)
\(\chi_{833}(718,\cdot)\)
\(\chi_{833}(744,\cdot)\)
\(\chi_{833}(786,\cdot)\)
\(\chi_{833}(795,\cdot)\)
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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 ]
\((52,785)\) → \((e\left(\frac{5}{21}\right),-i)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 833 }(4, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{31}{84}\right)\) |
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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)
