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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(508, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([63,55]))
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pari:[g,chi] = znchar(Mod(319,508))
| Modulus: | \(508\) | |
| Conductor: | \(508\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(126\) |
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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)
|
\(\chi_{508}(3,\cdot)\)
\(\chi_{508}(7,\cdot)\)
\(\chi_{508}(23,\cdot)\)
\(\chi_{508}(39,\cdot)\)
\(\chi_{508}(43,\cdot)\)
\(\chi_{508}(55,\cdot)\)
\(\chi_{508}(67,\cdot)\)
\(\chi_{508}(83,\cdot)\)
\(\chi_{508}(91,\cdot)\)
\(\chi_{508}(139,\cdot)\)
\(\chi_{508}(175,\cdot)\)
\(\chi_{508}(183,\cdot)\)
\(\chi_{508}(219,\cdot)\)
\(\chi_{508}(223,\cdot)\)
\(\chi_{508}(239,\cdot)\)
\(\chi_{508}(243,\cdot)\)
\(\chi_{508}(283,\cdot)\)
\(\chi_{508}(299,\cdot)\)
\(\chi_{508}(307,\cdot)\)
\(\chi_{508}(311,\cdot)\)
\(\chi_{508}(319,\cdot)\)
\(\chi_{508}(339,\cdot)\)
\(\chi_{508}(347,\cdot)\)
\(\chi_{508}(351,\cdot)\)
\(\chi_{508}(355,\cdot)\)
\(\chi_{508}(363,\cdot)\)
\(\chi_{508}(387,\cdot)\)
\(\chi_{508}(395,\cdot)\)
\(\chi_{508}(427,\cdot)\)
\(\chi_{508}(439,\cdot)\)
...
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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 ]
\((255,257)\) → \((-1,e\left(\frac{55}{126}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 508 }(319, a) \) |
\(1\) | \(1\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{23}{126}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{115}{126}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{40}{63}\right)\) |
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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)
