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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5417, base_ring=CyclotomicField(1354))
M = H._module
chi = DirichletCharacter(H, M([1169]))
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gp:[g,chi] = znchar(Mod(51, 5417))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5417.51");
| Modulus: | \(5417\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5417\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1354\) |
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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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\(\chi_{5417}(4,\cdot)\)
\(\chi_{5417}(11,\cdot)\)
\(\chi_{5417}(18,\cdot)\)
\(\chi_{5417}(29,\cdot)\)
\(\chi_{5417}(35,\cdot)\)
\(\chi_{5417}(47,\cdot)\)
\(\chi_{5417}(51,\cdot)\)
\(\chi_{5417}(60,\cdot)\)
\(\chi_{5417}(61,\cdot)\)
\(\chi_{5417}(64,\cdot)\)
\(\chi_{5417}(81,\cdot)\)
\(\chi_{5417}(82,\cdot)\)
\(\chi_{5417}(93,\cdot)\)
\(\chi_{5417}(98,\cdot)\)
\(\chi_{5417}(104,\cdot)\)
\(\chi_{5417}(106,\cdot)\)
\(\chi_{5417}(111,\cdot)\)
\(\chi_{5417}(113,\cdot)\)
\(\chi_{5417}(118,\cdot)\)
\(\chi_{5417}(133,\cdot)\)
\(\chi_{5417}(165,\cdot)\)
\(\chi_{5417}(168,\cdot)\)
\(\chi_{5417}(169,\cdot)\)
\(\chi_{5417}(170,\cdot)\)
\(\chi_{5417}(176,\cdot)\)
\(\chi_{5417}(181,\cdot)\)
\(\chi_{5417}(184,\cdot)\)
\(\chi_{5417}(200,\cdot)\)
\(\chi_{5417}(227,\cdot)\)
\(\chi_{5417}(228,\cdot)\)
...
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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 };
\(3\) → \(e\left(\frac{1169}{1354}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 5417 }(51, a) \) |
\(1\) | \(1\) | \(e\left(\frac{512}{677}\right)\) | \(e\left(\frac{1169}{1354}\right)\) | \(e\left(\frac{347}{677}\right)\) | \(e\left(\frac{261}{1354}\right)\) | \(e\left(\frac{839}{1354}\right)\) | \(e\left(\frac{999}{1354}\right)\) | \(e\left(\frac{182}{677}\right)\) | \(e\left(\frac{492}{677}\right)\) | \(e\left(\frac{1285}{1354}\right)\) | \(e\left(\frac{112}{677}\right)\) |
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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)
