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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(731, base_ring=CyclotomicField(336))
M = H._module
chi = DirichletCharacter(H, M([189,256]))
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gp:[g,chi] = znchar(Mod(99, 731))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("731.99");
| Modulus: | \(731\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(731\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(336\) |
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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);
|
\(\chi_{731}(10,\cdot)\)
\(\chi_{731}(14,\cdot)\)
\(\chi_{731}(23,\cdot)\)
\(\chi_{731}(24,\cdot)\)
\(\chi_{731}(31,\cdot)\)
\(\chi_{731}(40,\cdot)\)
\(\chi_{731}(56,\cdot)\)
\(\chi_{731}(57,\cdot)\)
\(\chi_{731}(58,\cdot)\)
\(\chi_{731}(74,\cdot)\)
\(\chi_{731}(95,\cdot)\)
\(\chi_{731}(96,\cdot)\)
\(\chi_{731}(99,\cdot)\)
\(\chi_{731}(109,\cdot)\)
\(\chi_{731}(124,\cdot)\)
\(\chi_{731}(126,\cdot)\)
\(\chi_{731}(139,\cdot)\)
\(\chi_{731}(142,\cdot)\)
\(\chi_{731}(143,\cdot)\)
\(\chi_{731}(146,\cdot)\)
\(\chi_{731}(160,\cdot)\)
\(\chi_{731}(167,\cdot)\)
\(\chi_{731}(181,\cdot)\)
\(\chi_{731}(182,\cdot)\)
\(\chi_{731}(197,\cdot)\)
\(\chi_{731}(210,\cdot)\)
\(\chi_{731}(224,\cdot)\)
\(\chi_{731}(228,\cdot)\)
\(\chi_{731}(232,\cdot)\)
\(\chi_{731}(267,\cdot)\)
...
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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 };
\((173,562)\) → \((e\left(\frac{9}{16}\right),e\left(\frac{16}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 731 }(99, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{109}{336}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{289}{336}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{109}{168}\right)\) | \(e\left(\frac{103}{336}\right)\) | \(e\left(\frac{89}{112}\right)\) |
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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)
