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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6503, base_ring=CyclotomicField(1392))
M = H._module
chi = DirichletCharacter(H, M([232,1047]))
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gp:[g,chi] = znchar(Mod(129, 6503))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6503.129");
| 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: | \(1392\) |
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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);
|
\(\chi_{6503}(10,\cdot)\)
\(\chi_{6503}(38,\cdot)\)
\(\chi_{6503}(45,\cdot)\)
\(\chi_{6503}(47,\cdot)\)
\(\chi_{6503}(61,\cdot)\)
\(\chi_{6503}(129,\cdot)\)
\(\chi_{6503}(145,\cdot)\)
\(\chi_{6503}(152,\cdot)\)
\(\chi_{6503}(157,\cdot)\)
\(\chi_{6503}(171,\cdot)\)
\(\chi_{6503}(178,\cdot)\)
\(\chi_{6503}(180,\cdot)\)
\(\chi_{6503}(206,\cdot)\)
\(\chi_{6503}(220,\cdot)\)
\(\chi_{6503}(227,\cdot)\)
\(\chi_{6503}(250,\cdot)\)
\(\chi_{6503}(255,\cdot)\)
\(\chi_{6503}(262,\cdot)\)
\(\chi_{6503}(332,\cdot)\)
\(\chi_{6503}(334,\cdot)\)
\(\chi_{6503}(346,\cdot)\)
\(\chi_{6503}(353,\cdot)\)
\(\chi_{6503}(355,\cdot)\)
\(\chi_{6503}(381,\cdot)\)
\(\chi_{6503}(402,\cdot)\)
\(\chi_{6503}(404,\cdot)\)
\(\chi_{6503}(446,\cdot)\)
\(\chi_{6503}(453,\cdot)\)
\(\chi_{6503}(465,\cdot)\)
\(\chi_{6503}(467,\cdot)\)
...
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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 };
\((5575,932)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{349}{464}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 6503 }(129, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{403}{696}\right)\) | \(e\left(\frac{1279}{1392}\right)\) | \(e\left(\frac{55}{348}\right)\) | \(e\left(\frac{251}{348}\right)\) | \(e\left(\frac{231}{464}\right)\) | \(e\left(\frac{171}{232}\right)\) | \(e\left(\frac{583}{696}\right)\) | \(e\left(\frac{209}{696}\right)\) | \(e\left(\frac{329}{696}\right)\) | \(e\left(\frac{107}{1392}\right)\) |
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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)
