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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(11243, base_ring=CyclotomicField(1606))
M = H._module
chi = DirichletCharacter(H, M([519]))
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gp:[g,chi] = znchar(Mod(2, 11243))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("11243.2");
| Modulus: | \(11243\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(11243\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1606\) |
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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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\(\chi_{11243}(2,\cdot)\)
\(\chi_{11243}(6,\cdot)\)
\(\chi_{11243}(8,\cdot)\)
\(\chi_{11243}(17,\cdot)\)
\(\chi_{11243}(18,\cdot)\)
\(\chi_{11243}(24,\cdot)\)
\(\chi_{11243}(32,\cdot)\)
\(\chi_{11243}(51,\cdot)\)
\(\chi_{11243}(54,\cdot)\)
\(\chi_{11243}(62,\cdot)\)
\(\chi_{11243}(67,\cdot)\)
\(\chi_{11243}(68,\cdot)\)
\(\chi_{11243}(72,\cdot)\)
\(\chi_{11243}(96,\cdot)\)
\(\chi_{11243}(128,\cdot)\)
\(\chi_{11243}(153,\cdot)\)
\(\chi_{11243}(162,\cdot)\)
\(\chi_{11243}(167,\cdot)\)
\(\chi_{11243}(186,\cdot)\)
\(\chi_{11243}(201,\cdot)\)
\(\chi_{11243}(204,\cdot)\)
\(\chi_{11243}(215,\cdot)\)
\(\chi_{11243}(216,\cdot)\)
\(\chi_{11243}(227,\cdot)\)
\(\chi_{11243}(248,\cdot)\)
\(\chi_{11243}(268,\cdot)\)
\(\chi_{11243}(272,\cdot)\)
\(\chi_{11243}(278,\cdot)\)
\(\chi_{11243}(288,\cdot)\)
\(\chi_{11243}(337,\cdot)\)
...
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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 };
\(5\) → \(e\left(\frac{519}{1606}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 11243 }(2, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{83}{1606}\right)\) | \(e\left(\frac{53}{73}\right)\) | \(e\left(\frac{83}{803}\right)\) | \(e\left(\frac{519}{1606}\right)\) | \(e\left(\frac{1249}{1606}\right)\) | \(e\left(\frac{1279}{1606}\right)\) | \(e\left(\frac{249}{1606}\right)\) | \(e\left(\frac{33}{73}\right)\) | \(e\left(\frac{301}{803}\right)\) | \(e\left(\frac{9}{1606}\right)\) |
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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)
