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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(192717, base_ring=CyclotomicField(1386))
M = H._module
chi = DirichletCharacter(H, M([231,132,385,1134]))
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gp:[g,chi] = znchar(Mod(5618, 192717))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("192717.5618");
| Modulus: | \(192717\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(192717\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1386\) |
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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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\(\chi_{192717}(200,\cdot)\)
\(\chi_{192717}(830,\cdot)\)
\(\chi_{192717}(1409,\cdot)\)
\(\chi_{192717}(1649,\cdot)\)
\(\chi_{192717}(2594,\cdot)\)
\(\chi_{192717}(2795,\cdot)\)
\(\chi_{192717}(3224,\cdot)\)
\(\chi_{192717}(3992,\cdot)\)
\(\chi_{192717}(4043,\cdot)\)
\(\chi_{192717}(5000,\cdot)\)
\(\chi_{192717}(5315,\cdot)\)
\(\chi_{192717}(5618,\cdot)\)
\(\chi_{192717}(6512,\cdot)\)
\(\chi_{192717}(7583,\cdot)\)
\(\chi_{192717}(7709,\cdot)\)
\(\chi_{192717}(8012,\cdot)\)
\(\chi_{192717}(8591,\cdot)\)
\(\chi_{192717}(9209,\cdot)\)
\(\chi_{192717}(9788,\cdot)\)
\(\chi_{192717}(10103,\cdot)\)
\(\chi_{192717}(10973,\cdot)\)
\(\chi_{192717}(12422,\cdot)\)
\(\chi_{192717}(12497,\cdot)\)
\(\chi_{192717}(12800,\cdot)\)
\(\chi_{192717}(13367,\cdot)\)
\(\chi_{192717}(13379,\cdot)\)
\(\chi_{192717}(13619,\cdot)\)
\(\chi_{192717}(13694,\cdot)\)
\(\chi_{192717}(13997,\cdot)\)
\(\chi_{192717}(15761,\cdot)\)
...
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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 };
\((107066,74728,60859,142444)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{2}{21}\right),e\left(\frac{5}{18}\right),e\left(\frac{9}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(20\) |
| \( \chi_{ 192717 }(5618, a) \) |
\(1\) | \(1\) | \(e\left(\frac{386}{693}\right)\) | \(e\left(\frac{79}{693}\right)\) | \(e\left(\frac{1189}{1386}\right)\) | \(e\left(\frac{155}{231}\right)\) | \(e\left(\frac{575}{1386}\right)\) | \(e\left(\frac{311}{462}\right)\) | \(e\left(\frac{443}{1386}\right)\) | \(e\left(\frac{158}{693}\right)\) | \(e\left(\frac{535}{1386}\right)\) | \(e\left(\frac{449}{462}\right)\) |
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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)
