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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5135, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([117,91,120]))
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gp:[g,chi] = znchar(Mod(1168, 5135))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5135.1168");
| Modulus: | \(5135\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5135\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(156\) |
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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);
|
\(\chi_{5135}(67,\cdot)\)
\(\chi_{5135}(97,\cdot)\)
\(\chi_{5135}(258,\cdot)\)
\(\chi_{5135}(457,\cdot)\)
\(\chi_{5135}(617,\cdot)\)
\(\chi_{5135}(618,\cdot)\)
\(\chi_{5135}(778,\cdot)\)
\(\chi_{5135}(812,\cdot)\)
\(\chi_{5135}(877,\cdot)\)
\(\chi_{5135}(1073,\cdot)\)
\(\chi_{5135}(1168,\cdot)\)
\(\chi_{5135}(1203,\cdot)\)
\(\chi_{5135}(1237,\cdot)\)
\(\chi_{5135}(1302,\cdot)\)
\(\chi_{5135}(1432,\cdot)\)
\(\chi_{5135}(1723,\cdot)\)
\(\chi_{5135}(1917,\cdot)\)
\(\chi_{5135}(1918,\cdot)\)
\(\chi_{5135}(1948,\cdot)\)
\(\chi_{5135}(1983,\cdot)\)
\(\chi_{5135}(2013,\cdot)\)
\(\chi_{5135}(2143,\cdot)\)
\(\chi_{5135}(2277,\cdot)\)
\(\chi_{5135}(2437,\cdot)\)
\(\chi_{5135}(2732,\cdot)\)
\(\chi_{5135}(2827,\cdot)\)
\(\chi_{5135}(2862,\cdot)\)
\(\chi_{5135}(2988,\cdot)\)
\(\chi_{5135}(3023,\cdot)\)
\(\chi_{5135}(3382,\cdot)\)
...
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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 };
\((3082,3161,2926)\) → \((-i,e\left(\frac{7}{12}\right),e\left(\frac{10}{13}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 5135 }(1168, a) \) |
\(1\) | \(1\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{55}{156}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{119}{156}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{61}{156}\right)\) | \(e\left(\frac{9}{52}\right)\) | \(e\left(\frac{9}{26}\right)\) |
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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)
