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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(24871, base_ring=CyclotomicField(240))
M = H._module
chi = DirichletCharacter(H, M([200,24,165,160]))
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gp:[g,chi] = znchar(Mod(10309, 24871))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("24871.10309");
| Modulus: | \(24871\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(24871\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(240\) |
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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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\(\chi_{24871}(600,\cdot)\)
\(\chi_{24871}(843,\cdot)\)
\(\chi_{24871}(976,\cdot)\)
\(\chi_{24871}(2063,\cdot)\)
\(\chi_{24871}(2173,\cdot)\)
\(\chi_{24871}(2196,\cdot)\)
\(\chi_{24871}(2306,\cdot)\)
\(\chi_{24871}(3104,\cdot)\)
\(\chi_{24871}(3526,\cdot)\)
\(\chi_{24871}(3769,\cdot)\)
\(\chi_{24871}(4457,\cdot)\)
\(\chi_{24871}(4567,\cdot)\)
\(\chi_{24871}(5122,\cdot)\)
\(\chi_{24871}(5365,\cdot)\)
\(\chi_{24871}(6030,\cdot)\)
\(\chi_{24871}(6585,\cdot)\)
\(\chi_{24871}(6695,\cdot)\)
\(\chi_{24871}(6718,\cdot)\)
\(\chi_{24871}(6828,\cdot)\)
\(\chi_{24871}(7383,\cdot)\)
\(\chi_{24871}(7915,\cdot)\)
\(\chi_{24871}(8048,\cdot)\)
\(\chi_{24871}(8291,\cdot)\)
\(\chi_{24871}(8846,\cdot)\)
\(\chi_{24871}(8956,\cdot)\)
\(\chi_{24871}(9378,\cdot)\)
\(\chi_{24871}(9644,\cdot)\)
\(\chi_{24871}(10309,\cdot)\)
\(\chi_{24871}(10841,\cdot)\)
\(\chi_{24871}(10951,\cdot)\)
...
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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 };
\((14213,4523,2927,11782)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{1}{10}\right),e\left(\frac{11}{16}\right),e\left(\frac{2}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 24871 }(10309, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{7}{120}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{161}{240}\right)\) | \(e\left(\frac{11}{240}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{41}{60}\right)\) |
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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)
