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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(967, base_ring=CyclotomicField(322))
M = H._module
chi = DirichletCharacter(H, M([129]))
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gp:[g,chi] = znchar(Mod(76, 967))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("967.76");
| Modulus: | \(967\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(967\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(322\) |
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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);
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\(\chi_{967}(3,\cdot)\)
\(\chi_{967}(10,\cdot)\)
\(\chi_{967}(23,\cdot)\)
\(\chi_{967}(24,\cdot)\)
\(\chi_{967}(26,\cdot)\)
\(\chi_{967}(27,\cdot)\)
\(\chi_{967}(29,\cdot)\)
\(\chi_{967}(33,\cdot)\)
\(\chi_{967}(67,\cdot)\)
\(\chi_{967}(76,\cdot)\)
\(\chi_{967}(80,\cdot)\)
\(\chi_{967}(90,\cdot)\)
\(\chi_{967}(109,\cdot)\)
\(\chi_{967}(110,\cdot)\)
\(\chi_{967}(112,\cdot)\)
\(\chi_{967}(125,\cdot)\)
\(\chi_{967}(126,\cdot)\)
\(\chi_{967}(154,\cdot)\)
\(\chi_{967}(158,\cdot)\)
\(\chi_{967}(167,\cdot)\)
\(\chi_{967}(170,\cdot)\)
\(\chi_{967}(178,\cdot)\)
\(\chi_{967}(184,\cdot)\)
\(\chi_{967}(186,\cdot)\)
\(\chi_{967}(191,\cdot)\)
\(\chi_{967}(192,\cdot)\)
\(\chi_{967}(207,\cdot)\)
\(\chi_{967}(208,\cdot)\)
\(\chi_{967}(213,\cdot)\)
\(\chi_{967}(214,\cdot)\)
...
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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 };
\(5\) → \(e\left(\frac{129}{322}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 967 }(76, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{134}{161}\right)\) | \(e\left(\frac{305}{322}\right)\) | \(e\left(\frac{107}{161}\right)\) | \(e\left(\frac{129}{322}\right)\) | \(e\left(\frac{251}{322}\right)\) | \(e\left(\frac{271}{322}\right)\) | \(e\left(\frac{80}{161}\right)\) | \(e\left(\frac{144}{161}\right)\) | \(e\left(\frac{75}{322}\right)\) | \(e\left(\frac{121}{161}\right)\) |
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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)
