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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5143, base_ring=CyclotomicField(138))
M = H._module
chi = DirichletCharacter(H, M([23,32]))
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gp:[g,chi] = znchar(Mod(2543, 5143))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5143.2543");
| Modulus: | \(5143\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5143\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(138\) |
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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: | even |
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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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\(\chi_{5143}(11,\cdot)\)
\(\chi_{5143}(159,\cdot)\)
\(\chi_{5143}(307,\cdot)\)
\(\chi_{5143}(344,\cdot)\)
\(\chi_{5143}(841,\cdot)\)
\(\chi_{5143}(862,\cdot)\)
\(\chi_{5143}(915,\cdot)\)
\(\chi_{5143}(1121,\cdot)\)
\(\chi_{5143}(1158,\cdot)\)
\(\chi_{5143}(1211,\cdot)\)
\(\chi_{5143}(1232,\cdot)\)
\(\chi_{5143}(1248,\cdot)\)
\(\chi_{5143}(1507,\cdot)\)
\(\chi_{5143}(1618,\cdot)\)
\(\chi_{5143}(1692,\cdot)\)
\(\chi_{5143}(1861,\cdot)\)
\(\chi_{5143}(1914,\cdot)\)
\(\chi_{5143}(1951,\cdot)\)
\(\chi_{5143}(2083,\cdot)\)
\(\chi_{5143}(2120,\cdot)\)
\(\chi_{5143}(2342,\cdot)\)
\(\chi_{5143}(2379,\cdot)\)
\(\chi_{5143}(2432,\cdot)\)
\(\chi_{5143}(2490,\cdot)\)
\(\chi_{5143}(2506,\cdot)\)
\(\chi_{5143}(2527,\cdot)\)
\(\chi_{5143}(2543,\cdot)\)
\(\chi_{5143}(2712,\cdot)\)
\(\chi_{5143}(2950,\cdot)\)
\(\chi_{5143}(3283,\cdot)\)
...
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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 };
\((557,4589)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{16}{69}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 5143 }(2543, a) \) |
\(1\) | \(1\) | \(e\left(\frac{55}{138}\right)\) | \(e\left(\frac{58}{69}\right)\) | \(e\left(\frac{55}{69}\right)\) | \(e\left(\frac{107}{138}\right)\) | \(e\left(\frac{11}{46}\right)\) | \(e\left(\frac{64}{69}\right)\) | \(e\left(\frac{9}{46}\right)\) | \(e\left(\frac{47}{69}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{43}{69}\right)\) |
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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)
