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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(33489, base_ring=CyclotomicField(366))
M = H._module
chi = DirichletCharacter(H, M([61,205]))
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gp:[g,chi] = znchar(Mod(5294, 33489))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("33489.5294");
| Modulus: | \(33489\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(33489\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(366\) |
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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: | odd |
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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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\(\chi_{33489}(14,\cdot)\)
\(\chi_{33489}(353,\cdot)\)
\(\chi_{33489}(563,\cdot)\)
\(\chi_{33489}(902,\cdot)\)
\(\chi_{33489}(1112,\cdot)\)
\(\chi_{33489}(1451,\cdot)\)
\(\chi_{33489}(2000,\cdot)\)
\(\chi_{33489}(2210,\cdot)\)
\(\chi_{33489}(2549,\cdot)\)
\(\chi_{33489}(2759,\cdot)\)
\(\chi_{33489}(3098,\cdot)\)
\(\chi_{33489}(3308,\cdot)\)
\(\chi_{33489}(3647,\cdot)\)
\(\chi_{33489}(3857,\cdot)\)
\(\chi_{33489}(4196,\cdot)\)
\(\chi_{33489}(4406,\cdot)\)
\(\chi_{33489}(4745,\cdot)\)
\(\chi_{33489}(4955,\cdot)\)
\(\chi_{33489}(5294,\cdot)\)
\(\chi_{33489}(5504,\cdot)\)
\(\chi_{33489}(5843,\cdot)\)
\(\chi_{33489}(6053,\cdot)\)
\(\chi_{33489}(6392,\cdot)\)
\(\chi_{33489}(6602,\cdot)\)
\(\chi_{33489}(6941,\cdot)\)
\(\chi_{33489}(7151,\cdot)\)
\(\chi_{33489}(7490,\cdot)\)
\(\chi_{33489}(7700,\cdot)\)
\(\chi_{33489}(8039,\cdot)\)
\(\chi_{33489}(8249,\cdot)\)
...
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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 };
\((26048,7444)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{205}{366}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 33489 }(5294, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{133}{183}\right)\) | \(e\left(\frac{83}{183}\right)\) | \(e\left(\frac{71}{122}\right)\) | \(e\left(\frac{221}{366}\right)\) | \(e\left(\frac{11}{61}\right)\) | \(e\left(\frac{113}{366}\right)\) | \(e\left(\frac{2}{183}\right)\) | \(e\left(\frac{53}{61}\right)\) | \(e\left(\frac{121}{366}\right)\) | \(e\left(\frac{166}{183}\right)\) |
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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)
