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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3905, base_ring=CyclotomicField(140))
M = H._module
chi = DirichletCharacter(H, M([35,112,30]))
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gp:[g,chi] = znchar(Mod(2082, 3905))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3905.2082");
| Modulus: | \(3905\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3905\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(140\) |
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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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\(\chi_{3905}(97,\cdot)\)
\(\chi_{3905}(168,\cdot)\)
\(\chi_{3905}(247,\cdot)\)
\(\chi_{3905}(323,\cdot)\)
\(\chi_{3905}(378,\cdot)\)
\(\chi_{3905}(467,\cdot)\)
\(\chi_{3905}(477,\cdot)\)
\(\chi_{3905}(548,\cdot)\)
\(\chi_{3905}(807,\cdot)\)
\(\chi_{3905}(878,\cdot)\)
\(\chi_{3905}(962,\cdot)\)
\(\chi_{3905}(1017,\cdot)\)
\(\chi_{3905}(1028,\cdot)\)
\(\chi_{3905}(1248,\cdot)\)
\(\chi_{3905}(1258,\cdot)\)
\(\chi_{3905}(1312,\cdot)\)
\(\chi_{3905}(1532,\cdot)\)
\(\chi_{3905}(1588,\cdot)\)
\(\chi_{3905}(1743,\cdot)\)
\(\chi_{3905}(1798,\cdot)\)
\(\chi_{3905}(1897,\cdot)\)
\(\chi_{3905}(2022,\cdot)\)
\(\chi_{3905}(2027,\cdot)\)
\(\chi_{3905}(2082,\cdot)\)
\(\chi_{3905}(2093,\cdot)\)
\(\chi_{3905}(2227,\cdot)\)
\(\chi_{3905}(2242,\cdot)\)
\(\chi_{3905}(2313,\cdot)\)
\(\chi_{3905}(2678,\cdot)\)
\(\chi_{3905}(2732,\cdot)\)
...
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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 };
\((782,1421,2421)\) → \((i,e\left(\frac{4}{5}\right),e\left(\frac{3}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 3905 }(2082, a) \) |
\(1\) | \(1\) | \(e\left(\frac{47}{140}\right)\) | \(e\left(\frac{101}{140}\right)\) | \(e\left(\frac{47}{70}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{9}{140}\right)\) | \(e\left(\frac{1}{140}\right)\) | \(e\left(\frac{31}{70}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{127}{140}\right)\) | \(e\left(\frac{2}{5}\right)\) |
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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)
