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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(479, base_ring=CyclotomicField(478))
M = H._module
chi = DirichletCharacter(H, M([115]))
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gp:[g,chi] = znchar(Mod(129, 479))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("479.129");
| Modulus: | \(479\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(479\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(478\) |
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sage:chi.multiplicative_order()
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gp:charorder(g,chi)
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magma:Order(chi);
|
| Real: | no |
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sage:chi.multiplicative_order() <= 2
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gp:charorder(g,chi) <= 2
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magma:Order(chi) le 2;
|
| Primitive: | yes |
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sage:chi.is_primitive()
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gp:#znconreyconductor(g,chi)==1
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magma:IsPrimitive(chi);
|
| Minimal: | yes |
| Parity: | odd |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{479}(13,\cdot)\)
\(\chi_{479}(17,\cdot)\)
\(\chi_{479}(19,\cdot)\)
\(\chi_{479}(26,\cdot)\)
\(\chi_{479}(29,\cdot)\)
\(\chi_{479}(31,\cdot)\)
\(\chi_{479}(34,\cdot)\)
\(\chi_{479}(37,\cdot)\)
\(\chi_{479}(38,\cdot)\)
\(\chi_{479}(39,\cdot)\)
\(\chi_{479}(41,\cdot)\)
\(\chi_{479}(43,\cdot)\)
\(\chi_{479}(47,\cdot)\)
\(\chi_{479}(51,\cdot)\)
\(\chi_{479}(52,\cdot)\)
\(\chi_{479}(53,\cdot)\)
\(\chi_{479}(57,\cdot)\)
\(\chi_{479}(58,\cdot)\)
\(\chi_{479}(59,\cdot)\)
\(\chi_{479}(62,\cdot)\)
\(\chi_{479}(65,\cdot)\)
\(\chi_{479}(67,\cdot)\)
\(\chi_{479}(68,\cdot)\)
\(\chi_{479}(74,\cdot)\)
\(\chi_{479}(76,\cdot)\)
\(\chi_{479}(78,\cdot)\)
\(\chi_{479}(79,\cdot)\)
\(\chi_{479}(82,\cdot)\)
\(\chi_{479}(83,\cdot)\)
\(\chi_{479}(85,\cdot)\)
...
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sage:chi.galois_orbit()
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gp:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
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magma:order := Order(chi);
{ chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
\(13\) → \(e\left(\frac{115}{478}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 479 }(129, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{22}{239}\right)\) | \(e\left(\frac{72}{239}\right)\) | \(e\left(\frac{44}{239}\right)\) | \(e\left(\frac{57}{239}\right)\) | \(e\left(\frac{94}{239}\right)\) | \(e\left(\frac{143}{239}\right)\) | \(e\left(\frac{66}{239}\right)\) | \(e\left(\frac{144}{239}\right)\) | \(e\left(\frac{79}{239}\right)\) | \(e\left(\frac{126}{239}\right)\) |
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sage:chi(x)
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gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
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magma:chi(x)
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sage:chi.gauss_sum(a)
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gp:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
