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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(840889, base_ring=CyclotomicField(3406))
M = H._module
chi = DirichletCharacter(H, M([1703,984]))
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gp:[g,chi] = znchar(Mod(13425, 840889))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("840889.13425");
| Modulus: | \(840889\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(120127\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(3406\) |
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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: | no, induced from \(\chi_{120127}(13425,\cdot)\) |
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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: | odd |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{840889}(244,\cdot)\)
\(\chi_{840889}(342,\cdot)\)
\(\chi_{840889}(587,\cdot)\)
\(\chi_{840889}(636,\cdot)\)
\(\chi_{840889}(979,\cdot)\)
\(\chi_{840889}(1224,\cdot)\)
\(\chi_{840889}(1763,\cdot)\)
\(\chi_{840889}(2596,\cdot)\)
\(\chi_{840889}(3576,\cdot)\)
\(\chi_{840889}(4899,\cdot)\)
\(\chi_{840889}(5193,\cdot)\)
\(\chi_{840889}(5732,\cdot)\)
\(\chi_{840889}(6663,\cdot)\)
\(\chi_{840889}(6761,\cdot)\)
\(\chi_{840889}(7006,\cdot)\)
\(\chi_{840889}(7055,\cdot)\)
\(\chi_{840889}(7398,\cdot)\)
\(\chi_{840889}(7643,\cdot)\)
\(\chi_{840889}(8182,\cdot)\)
\(\chi_{840889}(9015,\cdot)\)
\(\chi_{840889}(9995,\cdot)\)
\(\chi_{840889}(11318,\cdot)\)
\(\chi_{840889}(11612,\cdot)\)
\(\chi_{840889}(12151,\cdot)\)
\(\chi_{840889}(13082,\cdot)\)
\(\chi_{840889}(13180,\cdot)\)
\(\chi_{840889}(13425,\cdot)\)
\(\chi_{840889}(13474,\cdot)\)
\(\chi_{840889}(13817,\cdot)\)
\(\chi_{840889}(14062,\cdot)\)
...
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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 };
\((308899,686442)\) → \((-1,e\left(\frac{492}{1703}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 840889 }(13425, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{492}{1703}\right)\) | \(e\left(\frac{1883}{3406}\right)\) | \(e\left(\frac{984}{1703}\right)\) | \(e\left(\frac{3209}{3406}\right)\) | \(e\left(\frac{2867}{3406}\right)\) | \(e\left(\frac{1476}{1703}\right)\) | \(e\left(\frac{180}{1703}\right)\) | \(e\left(\frac{787}{3406}\right)\) | \(e\left(\frac{785}{1703}\right)\) | \(e\left(\frac{445}{3406}\right)\) |
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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)
