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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(43725, base_ring=CyclotomicField(260))
M = H._module
chi = DirichletCharacter(H, M([130,91,182,175]))
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gp:[g,chi] = znchar(Mod(11678, 43725))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("43725.11678");
| Modulus: | \(43725\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(43725\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(260\) |
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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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\(\chi_{43725}(2,\cdot)\)
\(\chi_{43725}(8,\cdot)\)
\(\chi_{43725}(128,\cdot)\)
\(\chi_{43725}(398,\cdot)\)
\(\chi_{43725}(512,\cdot)\)
\(\chi_{43725}(827,\cdot)\)
\(\chi_{43725}(1592,\cdot)\)
\(\chi_{43725}(2048,\cdot)\)
\(\chi_{43725}(2417,\cdot)\)
\(\chi_{43725}(2477,\cdot)\)
\(\chi_{43725}(2987,\cdot)\)
\(\chi_{43725}(3308,\cdot)\)
\(\chi_{43725}(4067,\cdot)\)
\(\chi_{43725}(4523,\cdot)\)
\(\chi_{43725}(4538,\cdot)\)
\(\chi_{43725}(4637,\cdot)\)
\(\chi_{43725}(5462,\cdot)\)
\(\chi_{43725}(6497,\cdot)\)
\(\chi_{43725}(6728,\cdot)\)
\(\chi_{43725}(7553,\cdot)\)
\(\chi_{43725}(9203,\cdot)\)
\(\chi_{43725}(9797,\cdot)\)
\(\chi_{43725}(9908,\cdot)\)
\(\chi_{43725}(10733,\cdot)\)
\(\chi_{43725}(11138,\cdot)\)
\(\chi_{43725}(11678,\cdot)\)
\(\chi_{43725}(12062,\cdot)\)
\(\chi_{43725}(12383,\cdot)\)
\(\chi_{43725}(14027,\cdot)\)
\(\chi_{43725}(14438,\cdot)\)
...
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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 };
\((14576,41977,11926,18976)\) → \((-1,e\left(\frac{7}{20}\right),e\left(\frac{7}{10}\right),e\left(\frac{35}{52}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
| \( \chi_{ 43725 }(11678, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{130}\right)\) | \(e\left(\frac{29}{65}\right)\) | \(e\left(\frac{19}{260}\right)\) | \(e\left(\frac{87}{130}\right)\) | \(e\left(\frac{131}{260}\right)\) | \(e\left(\frac{77}{260}\right)\) | \(e\left(\frac{58}{65}\right)\) | \(e\left(\frac{21}{260}\right)\) | \(e\left(\frac{79}{260}\right)\) | \(e\left(\frac{3}{5}\right)\) |
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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)
