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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3261, base_ring=CyclotomicField(1086))
M = H._module
chi = DirichletCharacter(H, M([543,1003]))
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gp:[g,chi] = znchar(Mod(1193, 3261))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3261.1193");
| Modulus: | \(3261\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3261\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1086\) |
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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);
|
\(\chi_{3261}(14,\cdot)\)
\(\chi_{3261}(20,\cdot)\)
\(\chi_{3261}(29,\cdot)\)
\(\chi_{3261}(38,\cdot)\)
\(\chi_{3261}(44,\cdot)\)
\(\chi_{3261}(53,\cdot)\)
\(\chi_{3261}(59,\cdot)\)
\(\chi_{3261}(62,\cdot)\)
\(\chi_{3261}(74,\cdot)\)
\(\chi_{3261}(80,\cdot)\)
\(\chi_{3261}(89,\cdot)\)
\(\chi_{3261}(92,\cdot)\)
\(\chi_{3261}(104,\cdot)\)
\(\chi_{3261}(122,\cdot)\)
\(\chi_{3261}(131,\cdot)\)
\(\chi_{3261}(134,\cdot)\)
\(\chi_{3261}(149,\cdot)\)
\(\chi_{3261}(170,\cdot)\)
\(\chi_{3261}(176,\cdot)\)
\(\chi_{3261}(179,\cdot)\)
\(\chi_{3261}(188,\cdot)\)
\(\chi_{3261}(215,\cdot)\)
\(\chi_{3261}(221,\cdot)\)
\(\chi_{3261}(224,\cdot)\)
\(\chi_{3261}(233,\cdot)\)
\(\chi_{3261}(248,\cdot)\)
\(\chi_{3261}(251,\cdot)\)
\(\chi_{3261}(263,\cdot)\)
\(\chi_{3261}(281,\cdot)\)
\(\chi_{3261}(287,\cdot)\)
...
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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 };
\((1088,1090)\) → \((-1,e\left(\frac{1003}{1086}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 3261 }(1193, a) \) |
\(1\) | \(1\) | \(e\left(\frac{617}{1086}\right)\) | \(e\left(\frac{74}{543}\right)\) | \(e\left(\frac{65}{181}\right)\) | \(e\left(\frac{121}{362}\right)\) | \(e\left(\frac{255}{362}\right)\) | \(e\left(\frac{1007}{1086}\right)\) | \(e\left(\frac{20}{181}\right)\) | \(e\left(\frac{505}{1086}\right)\) | \(e\left(\frac{490}{543}\right)\) | \(e\left(\frac{148}{543}\right)\) |
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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)
