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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(116909, base_ring=CyclotomicField(1104))
M = H._module
chi = DirichletCharacter(H, M([460,483,24]))
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gp:[g,chi] = znchar(Mod(5128, 116909))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("116909.5128");
| Modulus: | \(116909\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(116909\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1104\) |
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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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\(\chi_{116909}(45,\cdot)\)
\(\chi_{116909}(505,\cdot)\)
\(\chi_{116909}(804,\cdot)\)
\(\chi_{116909}(873,\cdot)\)
\(\chi_{116909}(942,\cdot)\)
\(\chi_{116909}(1034,\cdot)\)
\(\chi_{116909}(1540,\cdot)\)
\(\chi_{116909}(2230,\cdot)\)
\(\chi_{116909}(2437,\cdot)\)
\(\chi_{116909}(2598,\cdot)\)
\(\chi_{116909}(3564,\cdot)\)
\(\chi_{116909}(3794,\cdot)\)
\(\chi_{116909}(4024,\cdot)\)
\(\chi_{116909}(4323,\cdot)\)
\(\chi_{116909}(4461,\cdot)\)
\(\chi_{116909}(5059,\cdot)\)
\(\chi_{116909}(5128,\cdot)\)
\(\chi_{116909}(5588,\cdot)\)
\(\chi_{116909}(5887,\cdot)\)
\(\chi_{116909}(5956,\cdot)\)
\(\chi_{116909}(6025,\cdot)\)
\(\chi_{116909}(6117,\cdot)\)
\(\chi_{116909}(6623,\cdot)\)
\(\chi_{116909}(7313,\cdot)\)
\(\chi_{116909}(7520,\cdot)\)
\(\chi_{116909}(7681,\cdot)\)
\(\chi_{116909}(8647,\cdot)\)
\(\chi_{116909}(8877,\cdot)\)
\(\chi_{116909}(9107,\cdot)\)
\(\chi_{116909}(9406,\cdot)\)
...
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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 };
\((35973,27509,34919)\) → \((e\left(\frac{5}{12}\right),e\left(\frac{7}{16}\right),e\left(\frac{1}{46}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 116909 }(5128, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{491}{552}\right)\) | \(e\left(\frac{499}{1104}\right)\) | \(e\left(\frac{215}{276}\right)\) | \(e\left(\frac{353}{368}\right)\) | \(e\left(\frac{377}{1104}\right)\) | \(e\left(\frac{221}{1104}\right)\) | \(e\left(\frac{123}{184}\right)\) | \(e\left(\frac{499}{552}\right)\) | \(e\left(\frac{937}{1104}\right)\) | \(e\left(\frac{817}{1104}\right)\) |
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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)
