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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(363, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([55,13]))
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pari:[g,chi] = znchar(Mod(206,363))
| Modulus: | \(363\) | |
| Conductor: | \(363\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(110\) |
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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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\(\chi_{363}(2,\cdot)\)
\(\chi_{363}(8,\cdot)\)
\(\chi_{363}(17,\cdot)\)
\(\chi_{363}(29,\cdot)\)
\(\chi_{363}(35,\cdot)\)
\(\chi_{363}(41,\cdot)\)
\(\chi_{363}(50,\cdot)\)
\(\chi_{363}(62,\cdot)\)
\(\chi_{363}(68,\cdot)\)
\(\chi_{363}(74,\cdot)\)
\(\chi_{363}(83,\cdot)\)
\(\chi_{363}(95,\cdot)\)
\(\chi_{363}(101,\cdot)\)
\(\chi_{363}(107,\cdot)\)
\(\chi_{363}(116,\cdot)\)
\(\chi_{363}(128,\cdot)\)
\(\chi_{363}(134,\cdot)\)
\(\chi_{363}(140,\cdot)\)
\(\chi_{363}(149,\cdot)\)
\(\chi_{363}(167,\cdot)\)
\(\chi_{363}(173,\cdot)\)
\(\chi_{363}(182,\cdot)\)
\(\chi_{363}(194,\cdot)\)
\(\chi_{363}(200,\cdot)\)
\(\chi_{363}(206,\cdot)\)
\(\chi_{363}(227,\cdot)\)
\(\chi_{363}(248,\cdot)\)
\(\chi_{363}(260,\cdot)\)
\(\chi_{363}(266,\cdot)\)
\(\chi_{363}(272,\cdot)\)
...
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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 ]
\((122,244)\) → \((-1,e\left(\frac{13}{110}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 363 }(206, a) \) |
\(1\) | \(1\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{16}{55}\right)\) |
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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)
