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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(174915, base_ring=CyclotomicField(858))
M = H._module
chi = DirichletCharacter(H, M([572,429,781,624]))
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gp:[g,chi] = znchar(Mod(17989, 174915))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("174915.17989");
| Modulus: | \(174915\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(174915\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(858\) |
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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);
|
\(\chi_{174915}(4,\cdot)\)
\(\chi_{174915}(439,\cdot)\)
\(\chi_{174915}(1024,\cdot)\)
\(\chi_{174915}(2194,\cdot)\)
\(\chi_{174915}(2929,\cdot)\)
\(\chi_{174915}(3364,\cdot)\)
\(\chi_{174915}(3514,\cdot)\)
\(\chi_{174915}(3949,\cdot)\)
\(\chi_{174915}(4534,\cdot)\)
\(\chi_{174915}(5119,\cdot)\)
\(\chi_{174915}(5269,\cdot)\)
\(\chi_{174915}(5854,\cdot)\)
\(\chi_{174915}(7024,\cdot)\)
\(\chi_{174915}(8194,\cdot)\)
\(\chi_{174915}(8629,\cdot)\)
\(\chi_{174915}(8779,\cdot)\)
\(\chi_{174915}(9364,\cdot)\)
\(\chi_{174915}(11554,\cdot)\)
\(\chi_{174915}(12139,\cdot)\)
\(\chi_{174915}(13459,\cdot)\)
\(\chi_{174915}(13894,\cdot)\)
\(\chi_{174915}(14479,\cdot)\)
\(\chi_{174915}(15649,\cdot)\)
\(\chi_{174915}(16384,\cdot)\)
\(\chi_{174915}(16819,\cdot)\)
\(\chi_{174915}(16969,\cdot)\)
\(\chi_{174915}(17404,\cdot)\)
\(\chi_{174915}(17989,\cdot)\)
\(\chi_{174915}(18574,\cdot)\)
\(\chi_{174915}(18724,\cdot)\)
...
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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 };
\((136046,69967,166636,83656)\) → \((e\left(\frac{2}{3}\right),-1,e\left(\frac{71}{78}\right),e\left(\frac{8}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
| \( \chi_{ 174915 }(17989, a) \) |
\(1\) | \(1\) | \(e\left(\frac{76}{143}\right)\) | \(e\left(\frac{9}{143}\right)\) | \(e\left(\frac{164}{429}\right)\) | \(e\left(\frac{85}{143}\right)\) | \(e\left(\frac{277}{286}\right)\) | \(e\left(\frac{392}{429}\right)\) | \(e\left(\frac{18}{143}\right)\) | \(e\left(\frac{419}{858}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(-1\) |
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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)
