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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(891, base_ring=CyclotomicField(270))
M = H._module
chi = DirichletCharacter(H, M([115,108]))
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pari:[g,chi] = znchar(Mod(5,891))
| Modulus: | \(891\) | |
| Conductor: | \(891\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(270\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
|
\(\chi_{891}(5,\cdot)\)
\(\chi_{891}(14,\cdot)\)
\(\chi_{891}(20,\cdot)\)
\(\chi_{891}(38,\cdot)\)
\(\chi_{891}(47,\cdot)\)
\(\chi_{891}(59,\cdot)\)
\(\chi_{891}(86,\cdot)\)
\(\chi_{891}(92,\cdot)\)
\(\chi_{891}(104,\cdot)\)
\(\chi_{891}(113,\cdot)\)
\(\chi_{891}(119,\cdot)\)
\(\chi_{891}(137,\cdot)\)
\(\chi_{891}(146,\cdot)\)
\(\chi_{891}(158,\cdot)\)
\(\chi_{891}(185,\cdot)\)
\(\chi_{891}(191,\cdot)\)
\(\chi_{891}(203,\cdot)\)
\(\chi_{891}(212,\cdot)\)
\(\chi_{891}(218,\cdot)\)
\(\chi_{891}(236,\cdot)\)
\(\chi_{891}(245,\cdot)\)
\(\chi_{891}(257,\cdot)\)
\(\chi_{891}(284,\cdot)\)
\(\chi_{891}(290,\cdot)\)
\(\chi_{891}(302,\cdot)\)
\(\chi_{891}(311,\cdot)\)
\(\chi_{891}(317,\cdot)\)
\(\chi_{891}(335,\cdot)\)
\(\chi_{891}(344,\cdot)\)
\(\chi_{891}(356,\cdot)\)
...
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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 ]
\((650,244)\) → \((e\left(\frac{23}{54}\right),e\left(\frac{2}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 891 }(5, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{223}{270}\right)\) | \(e\left(\frac{88}{135}\right)\) | \(e\left(\frac{107}{270}\right)\) | \(e\left(\frac{83}{135}\right)\) | \(e\left(\frac{43}{90}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{109}{135}\right)\) | \(e\left(\frac{119}{270}\right)\) | \(e\left(\frac{41}{135}\right)\) | \(e\left(\frac{59}{90}\right)\) |
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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)
