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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(923, base_ring=CyclotomicField(140))
M = H._module
chi = DirichletCharacter(H, M([105,18]))
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gp:[g,chi] = znchar(Mod(473, 923))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("923.473");
| Modulus: | \(923\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(923\) |
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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);
|
\(\chi_{923}(21,\cdot)\)
\(\chi_{923}(31,\cdot)\)
\(\chi_{923}(44,\cdot)\)
\(\chi_{923}(47,\cdot)\)
\(\chi_{923}(99,\cdot)\)
\(\chi_{923}(138,\cdot)\)
\(\chi_{923}(164,\cdot)\)
\(\chi_{923}(177,\cdot)\)
\(\chi_{923}(203,\cdot)\)
\(\chi_{923}(226,\cdot)\)
\(\chi_{923}(255,\cdot)\)
\(\chi_{923}(265,\cdot)\)
\(\chi_{923}(268,\cdot)\)
\(\chi_{923}(278,\cdot)\)
\(\chi_{923}(281,\cdot)\)
\(\chi_{923}(291,\cdot)\)
\(\chi_{923}(317,\cdot)\)
\(\chi_{923}(343,\cdot)\)
\(\chi_{923}(346,\cdot)\)
\(\chi_{923}(408,\cdot)\)
\(\chi_{923}(411,\cdot)\)
\(\chi_{923}(424,\cdot)\)
\(\chi_{923}(437,\cdot)\)
\(\chi_{923}(447,\cdot)\)
\(\chi_{923}(473,\cdot)\)
\(\chi_{923}(489,\cdot)\)
\(\chi_{923}(525,\cdot)\)
\(\chi_{923}(528,\cdot)\)
\(\chi_{923}(541,\cdot)\)
\(\chi_{923}(564,\cdot)\)
...
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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 };
\((782,859)\) → \((-i,e\left(\frac{9}{70}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 923 }(473, a) \) |
\(1\) | \(1\) | \(e\left(\frac{73}{140}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{3}{70}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{121}{140}\right)\) | \(e\left(\frac{53}{140}\right)\) | \(e\left(\frac{79}{140}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{61}{70}\right)\) | \(e\left(\frac{33}{140}\right)\) |
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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)
