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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(973, base_ring=CyclotomicField(138))
M = H._module
chi = DirichletCharacter(H, M([115,111]))
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gp:[g,chi] = znchar(Mod(411, 973))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("973.411");
| Modulus: | \(973\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(973\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(138\) |
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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);
|
\(\chi_{973}(10,\cdot)\)
\(\chi_{973}(33,\cdot)\)
\(\chi_{973}(59,\cdot)\)
\(\chi_{973}(75,\cdot)\)
\(\chi_{973}(82,\cdot)\)
\(\chi_{973}(87,\cdot)\)
\(\chi_{973}(94,\cdot)\)
\(\chi_{973}(103,\cdot)\)
\(\chi_{973}(166,\cdot)\)
\(\chi_{973}(178,\cdot)\)
\(\chi_{973}(187,\cdot)\)
\(\chi_{973}(199,\cdot)\)
\(\chi_{973}(201,\cdot)\)
\(\chi_{973}(213,\cdot)\)
\(\chi_{973}(215,\cdot)\)
\(\chi_{973}(234,\cdot)\)
\(\chi_{973}(292,\cdot)\)
\(\chi_{973}(311,\cdot)\)
\(\chi_{973}(353,\cdot)\)
\(\chi_{973}(360,\cdot)\)
\(\chi_{973}(362,\cdot)\)
\(\chi_{973}(381,\cdot)\)
\(\chi_{973}(383,\cdot)\)
\(\chi_{973}(411,\cdot)\)
\(\chi_{973}(425,\cdot)\)
\(\chi_{973}(444,\cdot)\)
\(\chi_{973}(465,\cdot)\)
\(\chi_{973}(479,\cdot)\)
\(\chi_{973}(493,\cdot)\)
\(\chi_{973}(570,\cdot)\)
...
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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 };
\((696,141)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{37}{46}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 973 }(411, a) \) |
\(1\) | \(1\) | \(e\left(\frac{65}{138}\right)\) | \(e\left(\frac{56}{69}\right)\) | \(e\left(\frac{65}{69}\right)\) | \(e\left(\frac{47}{138}\right)\) | \(e\left(\frac{13}{46}\right)\) | \(e\left(\frac{19}{46}\right)\) | \(e\left(\frac{43}{69}\right)\) | \(e\left(\frac{56}{69}\right)\) | \(e\left(\frac{32}{69}\right)\) | \(e\left(\frac{52}{69}\right)\) |
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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)
