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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5984, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([0,10,56,5]))
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gp:[g,chi] = znchar(Mod(2757, 5984))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5984.2757");
| Modulus: | \(5984\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5984\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(80\) |
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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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\(\chi_{5984}(61,\cdot)\)
\(\chi_{5984}(277,\cdot)\)
\(\chi_{5984}(381,\cdot)\)
\(\chi_{5984}(437,\cdot)\)
\(\chi_{5984}(589,\cdot)\)
\(\chi_{5984}(677,\cdot)\)
\(\chi_{5984}(821,\cdot)\)
\(\chi_{5984}(941,\cdot)\)
\(\chi_{5984}(981,\cdot)\)
\(\chi_{5984}(1469,\cdot)\)
\(\chi_{5984}(1525,\cdot)\)
\(\chi_{5984}(1669,\cdot)\)
\(\chi_{5984}(1909,\cdot)\)
\(\chi_{5984}(2213,\cdot)\)
\(\chi_{5984}(2613,\cdot)\)
\(\chi_{5984}(2757,\cdot)\)
\(\chi_{5984}(3845,\cdot)\)
\(\chi_{5984}(3869,\cdot)\)
\(\chi_{5984}(4397,\cdot)\)
\(\chi_{5984}(4413,\cdot)\)
\(\chi_{5984}(4485,\cdot)\)
\(\chi_{5984}(4749,\cdot)\)
\(\chi_{5984}(4941,\cdot)\)
\(\chi_{5984}(4957,\cdot)\)
\(\chi_{5984}(5029,\cdot)\)
\(\chi_{5984}(5277,\cdot)\)
\(\chi_{5984}(5293,\cdot)\)
\(\chi_{5984}(5485,\cdot)\)
\(\chi_{5984}(5573,\cdot)\)
\(\chi_{5984}(5717,\cdot)\)
...
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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 };
\((4863,2245,4897,1057)\) → \((1,e\left(\frac{1}{8}\right),e\left(\frac{7}{10}\right),e\left(\frac{1}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(19\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 5984 }(2757, a) \) |
\(1\) | \(1\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{19}{40}\right)\) |
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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)
