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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4465, base_ring=CyclotomicField(276))
M = H._module
chi = DirichletCharacter(H, M([207,230,120]))
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gp:[g,chi] = znchar(Mod(943, 4465))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4465.943");
| Modulus: | \(4465\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4465\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(276\) |
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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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\(\chi_{4465}(8,\cdot)\)
\(\chi_{4465}(12,\cdot)\)
\(\chi_{4465}(27,\cdot)\)
\(\chi_{4465}(103,\cdot)\)
\(\chi_{4465}(122,\cdot)\)
\(\chi_{4465}(183,\cdot)\)
\(\chi_{4465}(202,\cdot)\)
\(\chi_{4465}(388,\cdot)\)
\(\chi_{4465}(392,\cdot)\)
\(\chi_{4465}(487,\cdot)\)
\(\chi_{4465}(502,\cdot)\)
\(\chi_{4465}(578,\cdot)\)
\(\chi_{4465}(582,\cdot)\)
\(\chi_{4465}(692,\cdot)\)
\(\chi_{4465}(768,\cdot)\)
\(\chi_{4465}(848,\cdot)\)
\(\chi_{4465}(863,\cdot)\)
\(\chi_{4465}(867,\cdot)\)
\(\chi_{4465}(882,\cdot)\)
\(\chi_{4465}(943,\cdot)\)
\(\chi_{4465}(958,\cdot)\)
\(\chi_{4465}(977,\cdot)\)
\(\chi_{4465}(1038,\cdot)\)
\(\chi_{4465}(1152,\cdot)\)
\(\chi_{4465}(1228,\cdot)\)
\(\chi_{4465}(1243,\cdot)\)
\(\chi_{4465}(1247,\cdot)\)
\(\chi_{4465}(1323,\cdot)\)
\(\chi_{4465}(1418,\cdot)\)
\(\chi_{4465}(1437,\cdot)\)
...
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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 };
\((1787,2586,381)\) → \((-i,e\left(\frac{5}{6}\right),e\left(\frac{10}{23}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 4465 }(943, a) \) |
\(1\) | \(1\) | \(e\left(\frac{113}{276}\right)\) | \(e\left(\frac{215}{276}\right)\) | \(e\left(\frac{113}{138}\right)\) | \(e\left(\frac{13}{69}\right)\) | \(e\left(\frac{61}{92}\right)\) | \(e\left(\frac{21}{92}\right)\) | \(e\left(\frac{77}{138}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{55}{92}\right)\) | \(e\left(\frac{55}{276}\right)\) |
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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)
