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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3115, base_ring=CyclotomicField(88))
M = H._module
chi = DirichletCharacter(H, M([44,44,83]))
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gp:[g,chi] = znchar(Mod(419, 3115))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3115.419");
| Modulus: | \(3115\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3115\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(88\) |
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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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\(\chi_{3115}(104,\cdot)\)
\(\chi_{3115}(209,\cdot)\)
\(\chi_{3115}(244,\cdot)\)
\(\chi_{3115}(349,\cdot)\)
\(\chi_{3115}(384,\cdot)\)
\(\chi_{3115}(419,\cdot)\)
\(\chi_{3115}(594,\cdot)\)
\(\chi_{3115}(629,\cdot)\)
\(\chi_{3115}(664,\cdot)\)
\(\chi_{3115}(699,\cdot)\)
\(\chi_{3115}(804,\cdot)\)
\(\chi_{3115}(839,\cdot)\)
\(\chi_{3115}(909,\cdot)\)
\(\chi_{3115}(944,\cdot)\)
\(\chi_{3115}(1014,\cdot)\)
\(\chi_{3115}(1049,\cdot)\)
\(\chi_{3115}(1119,\cdot)\)
\(\chi_{3115}(1154,\cdot)\)
\(\chi_{3115}(1259,\cdot)\)
\(\chi_{3115}(1294,\cdot)\)
\(\chi_{3115}(1329,\cdot)\)
\(\chi_{3115}(1364,\cdot)\)
\(\chi_{3115}(1539,\cdot)\)
\(\chi_{3115}(1574,\cdot)\)
\(\chi_{3115}(1609,\cdot)\)
\(\chi_{3115}(1714,\cdot)\)
\(\chi_{3115}(1749,\cdot)\)
\(\chi_{3115}(1854,\cdot)\)
\(\chi_{3115}(2169,\cdot)\)
\(\chi_{3115}(2239,\cdot)\)
...
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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 };
\((1247,1781,1961)\) → \((-1,-1,e\left(\frac{83}{88}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 3115 }(419, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{83}{88}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{47}{88}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{61}{88}\right)\) | \(e\left(\frac{4}{11}\right)\) |
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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)
