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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(329, base_ring=CyclotomicField(138))
M = H._module
chi = DirichletCharacter(H, M([23,12]))
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gp:[g,chi] = znchar(Mod(108, 329))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("329.108");
| Modulus: | \(329\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(329\) |
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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: | odd |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{329}(3,\cdot)\)
\(\chi_{329}(12,\cdot)\)
\(\chi_{329}(17,\cdot)\)
\(\chi_{329}(24,\cdot)\)
\(\chi_{329}(54,\cdot)\)
\(\chi_{329}(59,\cdot)\)
\(\chi_{329}(61,\cdot)\)
\(\chi_{329}(68,\cdot)\)
\(\chi_{329}(75,\cdot)\)
\(\chi_{329}(89,\cdot)\)
\(\chi_{329}(96,\cdot)\)
\(\chi_{329}(101,\cdot)\)
\(\chi_{329}(103,\cdot)\)
\(\chi_{329}(108,\cdot)\)
\(\chi_{329}(110,\cdot)\)
\(\chi_{329}(115,\cdot)\)
\(\chi_{329}(122,\cdot)\)
\(\chi_{329}(131,\cdot)\)
\(\chi_{329}(136,\cdot)\)
\(\chi_{329}(143,\cdot)\)
\(\chi_{329}(145,\cdot)\)
\(\chi_{329}(150,\cdot)\)
\(\chi_{329}(157,\cdot)\)
\(\chi_{329}(159,\cdot)\)
\(\chi_{329}(166,\cdot)\)
\(\chi_{329}(173,\cdot)\)
\(\chi_{329}(178,\cdot)\)
\(\chi_{329}(192,\cdot)\)
\(\chi_{329}(194,\cdot)\)
\(\chi_{329}(206,\cdot)\)
...
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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 };
\((283,99)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{2}{23}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 329 }(108, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{62}{69}\right)\) | \(e\left(\frac{125}{138}\right)\) | \(e\left(\frac{55}{69}\right)\) | \(e\left(\frac{127}{138}\right)\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{56}{69}\right)\) | \(e\left(\frac{113}{138}\right)\) | \(e\left(\frac{19}{69}\right)\) | \(e\left(\frac{97}{138}\right)\) |
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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)
