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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2359, base_ring=CyclotomicField(168))
M = H._module
chi = DirichletCharacter(H, M([28,67]))
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gp:[g,chi] = znchar(Mod(829, 2359))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2359.829");
| Modulus: | \(2359\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2359\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(168\) |
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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);
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\(\chi_{2359}(3,\cdot)\)
\(\chi_{2359}(12,\cdot)\)
\(\chi_{2359}(24,\cdot)\)
\(\chi_{2359}(108,\cdot)\)
\(\chi_{2359}(192,\cdot)\)
\(\chi_{2359}(222,\cdot)\)
\(\chi_{2359}(243,\cdot)\)
\(\chi_{2359}(432,\cdot)\)
\(\chi_{2359}(437,\cdot)\)
\(\chi_{2359}(444,\cdot)\)
\(\chi_{2359}(486,\cdot)\)
\(\chi_{2359}(579,\cdot)\)
\(\chi_{2359}(768,\cdot)\)
\(\chi_{2359}(787,\cdot)\)
\(\chi_{2359}(789,\cdot)\)
\(\chi_{2359}(829,\cdot)\)
\(\chi_{2359}(864,\cdot)\)
\(\chi_{2359}(885,\cdot)\)
\(\chi_{2359}(983,\cdot)\)
\(\chi_{2359}(999,\cdot)\)
\(\chi_{2359}(1039,\cdot)\)
\(\chi_{2359}(1097,\cdot)\)
\(\chi_{2359}(1137,\cdot)\)
\(\chi_{2359}(1158,\cdot)\)
\(\chi_{2359}(1181,\cdot)\)
\(\chi_{2359}(1193,\cdot)\)
\(\chi_{2359}(1235,\cdot)\)
\(\chi_{2359}(1398,\cdot)\)
\(\chi_{2359}(1426,\cdot)\)
\(\chi_{2359}(1536,\cdot)\)
...
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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 };
\((675,1695)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{67}{168}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 2359 }(829, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{167}{168}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{95}{168}\right)\) | \(e\left(\frac{145}{168}\right)\) | \(e\left(\frac{15}{28}\right)\) |
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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)
