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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5587, base_ring=CyclotomicField(450))
M = H._module
chi = DirichletCharacter(H, M([25,306]))
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gp:[g,chi] = znchar(Mod(1077, 5587))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5587.1077");
| Modulus: | \(5587\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5587\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(450\) |
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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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\(\chi_{5587}(78,\cdot)\)
\(\chi_{5587}(299,\cdot)\)
\(\chi_{5587}(374,\cdot)\)
\(\chi_{5587}(400,\cdot)\)
\(\chi_{5587}(521,\cdot)\)
\(\chi_{5587}(539,\cdot)\)
\(\chi_{5587}(576,\cdot)\)
\(\chi_{5587}(580,\cdot)\)
\(\chi_{5587}(613,\cdot)\)
\(\chi_{5587}(633,\cdot)\)
\(\chi_{5587}(654,\cdot)\)
\(\chi_{5587}(728,\cdot)\)
\(\chi_{5587}(731,\cdot)\)
\(\chi_{5587}(805,\cdot)\)
\(\chi_{5587}(839,\cdot)\)
\(\chi_{5587}(879,\cdot)\)
\(\chi_{5587}(950,\cdot)\)
\(\chi_{5587}(987,\cdot)\)
\(\chi_{5587}(990,\cdot)\)
\(\chi_{5587}(992,\cdot)\)
\(\chi_{5587}(1029,\cdot)\)
\(\chi_{5587}(1066,\cdot)\)
\(\chi_{5587}(1077,\cdot)\)
\(\chi_{5587}(1101,\cdot)\)
\(\chi_{5587}(1135,\cdot)\)
\(\chi_{5587}(1138,\cdot)\)
\(\chi_{5587}(1151,\cdot)\)
\(\chi_{5587}(1205,\cdot)\)
\(\chi_{5587}(1286,\cdot)\)
\(\chi_{5587}(1299,\cdot)\)
...
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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 };
\((3776,3479)\) → \((e\left(\frac{1}{18}\right),e\left(\frac{17}{25}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 5587 }(1077, a) \) |
\(1\) | \(1\) | \(e\left(\frac{59}{90}\right)\) | \(e\left(\frac{118}{225}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{197}{450}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{76}{225}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{11}{225}\right)\) | \(e\left(\frac{7}{75}\right)\) | \(e\left(\frac{44}{75}\right)\) |
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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)
