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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6503, base_ring=CyclotomicField(2784))
M = H._module
chi = DirichletCharacter(H, M([928,921]))
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gp:[g,chi] = znchar(Mod(268, 6503))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6503.268");
| Modulus: | \(6503\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(6503\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(2784\) |
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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);
|
\(\chi_{6503}(30,\cdot)\)
\(\chi_{6503}(60,\cdot)\)
\(\chi_{6503}(65,\cdot)\)
\(\chi_{6503}(67,\cdot)\)
\(\chi_{6503}(79,\cdot)\)
\(\chi_{6503}(86,\cdot)\)
\(\chi_{6503}(107,\cdot)\)
\(\chi_{6503}(109,\cdot)\)
\(\chi_{6503}(114,\cdot)\)
\(\chi_{6503}(130,\cdot)\)
\(\chi_{6503}(135,\cdot)\)
\(\chi_{6503}(151,\cdot)\)
\(\chi_{6503}(158,\cdot)\)
\(\chi_{6503}(163,\cdot)\)
\(\chi_{6503}(165,\cdot)\)
\(\chi_{6503}(172,\cdot)\)
\(\chi_{6503}(179,\cdot)\)
\(\chi_{6503}(191,\cdot)\)
\(\chi_{6503}(205,\cdot)\)
\(\chi_{6503}(214,\cdot)\)
\(\chi_{6503}(233,\cdot)\)
\(\chi_{6503}(240,\cdot)\)
\(\chi_{6503}(247,\cdot)\)
\(\chi_{6503}(249,\cdot)\)
\(\chi_{6503}(254,\cdot)\)
\(\chi_{6503}(268,\cdot)\)
\(\chi_{6503}(270,\cdot)\)
\(\chi_{6503}(277,\cdot)\)
\(\chi_{6503}(282,\cdot)\)
\(\chi_{6503}(303,\cdot)\)
...
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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 };
\((5575,932)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{307}{928}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 6503 }(268, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{877}{1392}\right)\) | \(e\left(\frac{1849}{2784}\right)\) | \(e\left(\frac{181}{696}\right)\) | \(e\left(\frac{323}{696}\right)\) | \(e\left(\frac{273}{928}\right)\) | \(e\left(\frac{413}{464}\right)\) | \(e\left(\frac{457}{1392}\right)\) | \(e\left(\frac{131}{1392}\right)\) | \(e\left(\frac{431}{1392}\right)\) | \(e\left(\frac{2573}{2784}\right)\) |
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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)
