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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2497, base_ring=CyclotomicField(1130))
M = H._module
chi = DirichletCharacter(H, M([678,355]))
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gp:[g,chi] = znchar(Mod(1054, 2497))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2497.1054");
| Modulus: | \(2497\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2497\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1130\) |
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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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\(\chi_{2497}(5,\cdot)\)
\(\chi_{2497}(14,\cdot)\)
\(\chi_{2497}(15,\cdot)\)
\(\chi_{2497}(20,\cdot)\)
\(\chi_{2497}(31,\cdot)\)
\(\chi_{2497}(37,\cdot)\)
\(\chi_{2497}(38,\cdot)\)
\(\chi_{2497}(42,\cdot)\)
\(\chi_{2497}(58,\cdot)\)
\(\chi_{2497}(60,\cdot)\)
\(\chi_{2497}(80,\cdot)\)
\(\chi_{2497}(86,\cdot)\)
\(\chi_{2497}(91,\cdot)\)
\(\chi_{2497}(93,\cdot)\)
\(\chi_{2497}(114,\cdot)\)
\(\chi_{2497}(115,\cdot)\)
\(\chi_{2497}(119,\cdot)\)
\(\chi_{2497}(124,\cdot)\)
\(\chi_{2497}(125,\cdot)\)
\(\chi_{2497}(126,\cdot)\)
\(\chi_{2497}(130,\cdot)\)
\(\chi_{2497}(135,\cdot)\)
\(\chi_{2497}(137,\cdot)\)
\(\chi_{2497}(146,\cdot)\)
\(\chi_{2497}(148,\cdot)\)
\(\chi_{2497}(152,\cdot)\)
\(\chi_{2497}(157,\cdot)\)
\(\chi_{2497}(158,\cdot)\)
\(\chi_{2497}(163,\cdot)\)
\(\chi_{2497}(168,\cdot)\)
...
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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 };
\((1817,683)\) → \((e\left(\frac{3}{5}\right),e\left(\frac{71}{226}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 2497 }(1054, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1033}{1130}\right)\) | \(e\left(\frac{142}{565}\right)\) | \(e\left(\frac{468}{565}\right)\) | \(e\left(\frac{967}{1130}\right)\) | \(e\left(\frac{187}{1130}\right)\) | \(e\left(\frac{328}{565}\right)\) | \(e\left(\frac{839}{1130}\right)\) | \(e\left(\frac{284}{565}\right)\) | \(e\left(\frac{87}{113}\right)\) | \(e\left(\frac{9}{113}\right)\) |
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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)
