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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(725, base_ring=CyclotomicField(140))
M = H._module
chi = DirichletCharacter(H, M([77,45]))
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gp:[g,chi] = znchar(Mod(48, 725))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("725.48");
| Modulus: | \(725\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(725\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(140\) |
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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: | even |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{725}(3,\cdot)\)
\(\chi_{725}(27,\cdot)\)
\(\chi_{725}(37,\cdot)\)
\(\chi_{725}(47,\cdot)\)
\(\chi_{725}(48,\cdot)\)
\(\chi_{725}(97,\cdot)\)
\(\chi_{725}(98,\cdot)\)
\(\chi_{725}(102,\cdot)\)
\(\chi_{725}(108,\cdot)\)
\(\chi_{725}(142,\cdot)\)
\(\chi_{725}(148,\cdot)\)
\(\chi_{725}(172,\cdot)\)
\(\chi_{725}(188,\cdot)\)
\(\chi_{725}(192,\cdot)\)
\(\chi_{725}(242,\cdot)\)
\(\chi_{725}(247,\cdot)\)
\(\chi_{725}(253,\cdot)\)
\(\chi_{725}(263,\cdot)\)
\(\chi_{725}(287,\cdot)\)
\(\chi_{725}(317,\cdot)\)
\(\chi_{725}(327,\cdot)\)
\(\chi_{725}(333,\cdot)\)
\(\chi_{725}(337,\cdot)\)
\(\chi_{725}(338,\cdot)\)
\(\chi_{725}(387,\cdot)\)
\(\chi_{725}(388,\cdot)\)
\(\chi_{725}(392,\cdot)\)
\(\chi_{725}(398,\cdot)\)
\(\chi_{725}(408,\cdot)\)
\(\chi_{725}(438,\cdot)\)
...
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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 };
\((552,176)\) → \((e\left(\frac{11}{20}\right),e\left(\frac{9}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 725 }(48, a) \) |
\(1\) | \(1\) | \(e\left(\frac{61}{70}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{23}{70}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{43}{70}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{117}{140}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{33}{140}\right)\) |
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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)
