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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(835, base_ring=CyclotomicField(166))
M = H._module
chi = DirichletCharacter(H, M([83,85]))
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gp:[g,chi] = znchar(Mod(309, 835))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("835.309");
| Modulus: | \(835\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(835\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(166\) |
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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);
|
\(\chi_{835}(34,\cdot)\)
\(\chi_{835}(39,\cdot)\)
\(\chi_{835}(59,\cdot)\)
\(\chi_{835}(69,\cdot)\)
\(\chi_{835}(74,\cdot)\)
\(\chi_{835}(79,\cdot)\)
\(\chi_{835}(104,\cdot)\)
\(\chi_{835}(109,\cdot)\)
\(\chi_{835}(119,\cdot)\)
\(\chi_{835}(129,\cdot)\)
\(\chi_{835}(134,\cdot)\)
\(\chi_{835}(139,\cdot)\)
\(\chi_{835}(149,\cdot)\)
\(\chi_{835}(159,\cdot)\)
\(\chi_{835}(164,\cdot)\)
\(\chi_{835}(184,\cdot)\)
\(\chi_{835}(204,\cdot)\)
\(\chi_{835}(219,\cdot)\)
\(\chi_{835}(234,\cdot)\)
\(\chi_{835}(249,\cdot)\)
\(\chi_{835}(259,\cdot)\)
\(\chi_{835}(269,\cdot)\)
\(\chi_{835}(284,\cdot)\)
\(\chi_{835}(309,\cdot)\)
\(\chi_{835}(339,\cdot)\)
\(\chi_{835}(344,\cdot)\)
\(\chi_{835}(349,\cdot)\)
\(\chi_{835}(354,\cdot)\)
\(\chi_{835}(364,\cdot)\)
\(\chi_{835}(369,\cdot)\)
...
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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 };
\((502,506)\) → \((-1,e\left(\frac{85}{166}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 835 }(309, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{163}{166}\right)\) | \(e\left(\frac{105}{166}\right)\) | \(e\left(\frac{80}{83}\right)\) | \(e\left(\frac{51}{83}\right)\) | \(e\left(\frac{153}{166}\right)\) | \(e\left(\frac{157}{166}\right)\) | \(e\left(\frac{22}{83}\right)\) | \(e\left(\frac{28}{83}\right)\) | \(e\left(\frac{99}{166}\right)\) | \(e\left(\frac{20}{83}\right)\) |
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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)
