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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5117, base_ring=CyclotomicField(168))
M = H._module
chi = DirichletCharacter(H, M([140,63,76]))
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gp:[g,chi] = znchar(Mod(3330, 5117))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5117.3330");
| Modulus: | \(5117\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5117\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(168\) |
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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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\(\chi_{5117}(19,\cdot)\)
\(\chi_{5117}(206,\cdot)\)
\(\chi_{5117}(416,\cdot)\)
\(\chi_{5117}(542,\cdot)\)
\(\chi_{5117}(614,\cdot)\)
\(\chi_{5117}(621,\cdot)\)
\(\chi_{5117}(808,\cdot)\)
\(\chi_{5117}(1018,\cdot)\)
\(\chi_{5117}(1181,\cdot)\)
\(\chi_{5117}(1209,\cdot)\)
\(\chi_{5117}(1216,\cdot)\)
\(\chi_{5117}(1277,\cdot)\)
\(\chi_{5117}(1538,\cdot)\)
\(\chi_{5117}(1783,\cdot)\)
\(\chi_{5117}(1811,\cdot)\)
\(\chi_{5117}(1879,\cdot)\)
\(\chi_{5117}(1895,\cdot)\)
\(\chi_{5117}(1963,\cdot)\)
\(\chi_{5117}(2082,\cdot)\)
\(\chi_{5117}(2140,\cdot)\)
\(\chi_{5117}(2348,\cdot)\)
\(\chi_{5117}(2497,\cdot)\)
\(\chi_{5117}(2565,\cdot)\)
\(\chi_{5117}(2684,\cdot)\)
\(\chi_{5117}(2728,\cdot)\)
\(\chi_{5117}(2824,\cdot)\)
\(\chi_{5117}(2915,\cdot)\)
\(\chi_{5117}(2950,\cdot)\)
\(\chi_{5117}(3323,\cdot)\)
\(\chi_{5117}(3330,\cdot)\)
...
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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 };
\((2194,904,2024)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{3}{8}\right),e\left(\frac{19}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 5117 }(3330, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{59}{168}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{89}{168}\right)\) | \(e\left(\frac{155}{168}\right)\) |
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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)
