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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(108445, base_ring=CyclotomicField(184))
M = H._module
chi = DirichletCharacter(H, M([46,156,69]))
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gp:[g,chi] = znchar(Mod(3242, 108445))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("108445.3242");
| Modulus: | \(108445\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(108445\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(184\) |
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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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\(\chi_{108445}(68,\cdot)\)
\(\chi_{108445}(2023,\cdot)\)
\(\chi_{108445}(3242,\cdot)\)
\(\chi_{108445}(4507,\cdot)\)
\(\chi_{108445}(4783,\cdot)\)
\(\chi_{108445}(6738,\cdot)\)
\(\chi_{108445}(7957,\cdot)\)
\(\chi_{108445}(9222,\cdot)\)
\(\chi_{108445}(9498,\cdot)\)
\(\chi_{108445}(11453,\cdot)\)
\(\chi_{108445}(12672,\cdot)\)
\(\chi_{108445}(13937,\cdot)\)
\(\chi_{108445}(14213,\cdot)\)
\(\chi_{108445}(16168,\cdot)\)
\(\chi_{108445}(17387,\cdot)\)
\(\chi_{108445}(18652,\cdot)\)
\(\chi_{108445}(18928,\cdot)\)
\(\chi_{108445}(20883,\cdot)\)
\(\chi_{108445}(22102,\cdot)\)
\(\chi_{108445}(23367,\cdot)\)
\(\chi_{108445}(23643,\cdot)\)
\(\chi_{108445}(25598,\cdot)\)
\(\chi_{108445}(26817,\cdot)\)
\(\chi_{108445}(28082,\cdot)\)
\(\chi_{108445}(28358,\cdot)\)
\(\chi_{108445}(30313,\cdot)\)
\(\chi_{108445}(31532,\cdot)\)
\(\chi_{108445}(33073,\cdot)\)
\(\chi_{108445}(35028,\cdot)\)
\(\chi_{108445}(36247,\cdot)\)
...
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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 };
\((86757,65601,26451)\) → \((i,e\left(\frac{39}{46}\right),e\left(\frac{3}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 108445 }(3242, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{173}{184}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{93}{184}\right)\) | \(e\left(\frac{45}{184}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{81}{92}\right)\) | \(e\left(\frac{147}{184}\right)\) | \(e\left(\frac{13}{184}\right)\) | \(e\left(\frac{85}{184}\right)\) |
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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)
