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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(707, base_ring=CyclotomicField(300))
M = H._module
chi = DirichletCharacter(H, M([200,231]))
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gp:[g,chi] = znchar(Mod(263, 707))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("707.263");
| Modulus: | \(707\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(707\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(300\) |
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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);
|
\(\chi_{707}(2,\cdot)\)
\(\chi_{707}(11,\cdot)\)
\(\chi_{707}(18,\cdot)\)
\(\chi_{707}(46,\cdot)\)
\(\chi_{707}(51,\cdot)\)
\(\chi_{707}(53,\cdot)\)
\(\chi_{707}(67,\cdot)\)
\(\chi_{707}(72,\cdot)\)
\(\chi_{707}(74,\cdot)\)
\(\chi_{707}(86,\cdot)\)
\(\chi_{707}(93,\cdot)\)
\(\chi_{707}(109,\cdot)\)
\(\chi_{707}(116,\cdot)\)
\(\chi_{707}(128,\cdot)\)
\(\chi_{707}(130,\cdot)\)
\(\chi_{707}(135,\cdot)\)
\(\chi_{707}(149,\cdot)\)
\(\chi_{707}(151,\cdot)\)
\(\chi_{707}(156,\cdot)\)
\(\chi_{707}(184,\cdot)\)
\(\chi_{707}(191,\cdot)\)
\(\chi_{707}(200,\cdot)\)
\(\chi_{707}(205,\cdot)\)
\(\chi_{707}(214,\cdot)\)
\(\chi_{707}(228,\cdot)\)
\(\chi_{707}(240,\cdot)\)
\(\chi_{707}(242,\cdot)\)
\(\chi_{707}(261,\cdot)\)
\(\chi_{707}(263,\cdot)\)
\(\chi_{707}(268,\cdot)\)
...
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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 };
\((304,204)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{77}{100}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 707 }(263, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{31}{300}\right)\) | \(e\left(\frac{239}{300}\right)\) | \(e\left(\frac{31}{150}\right)\) | \(e\left(\frac{61}{75}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{31}{100}\right)\) | \(e\left(\frac{89}{150}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{203}{300}\right)\) | \(e\left(\frac{1}{300}\right)\) |
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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)
