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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(517, base_ring=CyclotomicField(230))
M = H._module
chi = DirichletCharacter(H, M([138,15]))
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pari:[g,chi] = znchar(Mod(31,517))
| Modulus: | \(517\) | |
| Conductor: | \(517\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(230\) |
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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)
|
\(\chi_{517}(5,\cdot)\)
\(\chi_{517}(15,\cdot)\)
\(\chi_{517}(20,\cdot)\)
\(\chi_{517}(26,\cdot)\)
\(\chi_{517}(31,\cdot)\)
\(\chi_{517}(38,\cdot)\)
\(\chi_{517}(58,\cdot)\)
\(\chi_{517}(60,\cdot)\)
\(\chi_{517}(69,\cdot)\)
\(\chi_{517}(70,\cdot)\)
\(\chi_{517}(80,\cdot)\)
\(\chi_{517}(82,\cdot)\)
\(\chi_{517}(86,\cdot)\)
\(\chi_{517}(91,\cdot)\)
\(\chi_{517}(92,\cdot)\)
\(\chi_{517}(104,\cdot)\)
\(\chi_{517}(113,\cdot)\)
\(\chi_{517}(114,\cdot)\)
\(\chi_{517}(124,\cdot)\)
\(\chi_{517}(125,\cdot)\)
\(\chi_{517}(135,\cdot)\)
\(\chi_{517}(137,\cdot)\)
\(\chi_{517}(146,\cdot)\)
\(\chi_{517}(152,\cdot)\)
\(\chi_{517}(163,\cdot)\)
\(\chi_{517}(170,\cdot)\)
\(\chi_{517}(174,\cdot)\)
\(\chi_{517}(179,\cdot)\)
\(\chi_{517}(180,\cdot)\)
\(\chi_{517}(181,\cdot)\)
...
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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 ]
\((189,287)\) → \((e\left(\frac{3}{5}\right),e\left(\frac{3}{46}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 517 }(31, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{89}{115}\right)\) | \(e\left(\frac{12}{115}\right)\) | \(e\left(\frac{63}{115}\right)\) | \(e\left(\frac{107}{230}\right)\) | \(e\left(\frac{101}{115}\right)\) | \(e\left(\frac{33}{115}\right)\) | \(e\left(\frac{37}{115}\right)\) | \(e\left(\frac{24}{115}\right)\) | \(e\left(\frac{11}{46}\right)\) | \(e\left(\frac{15}{23}\right)\) |
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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)
