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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3320, base_ring=CyclotomicField(164))
M = H._module
chi = DirichletCharacter(H, M([82,82,41,146]))
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gp:[g,chi] = znchar(Mod(587, 3320))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3320.587");
| Modulus: | \(3320\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3320\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(164\) |
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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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\(\chi_{3320}(43,\cdot)\)
\(\chi_{3320}(67,\cdot)\)
\(\chi_{3320}(107,\cdot)\)
\(\chi_{3320}(163,\cdot)\)
\(\chi_{3320}(267,\cdot)\)
\(\chi_{3320}(283,\cdot)\)
\(\chi_{3320}(307,\cdot)\)
\(\chi_{3320}(323,\cdot)\)
\(\chi_{3320}(347,\cdot)\)
\(\chi_{3320}(387,\cdot)\)
\(\chi_{3320}(403,\cdot)\)
\(\chi_{3320}(467,\cdot)\)
\(\chi_{3320}(587,\cdot)\)
\(\chi_{3320}(603,\cdot)\)
\(\chi_{3320}(627,\cdot)\)
\(\chi_{3320}(643,\cdot)\)
\(\chi_{3320}(683,\cdot)\)
\(\chi_{3320}(707,\cdot)\)
\(\chi_{3320}(803,\cdot)\)
\(\chi_{3320}(827,\cdot)\)
\(\chi_{3320}(843,\cdot)\)
\(\chi_{3320}(883,\cdot)\)
\(\chi_{3320}(947,\cdot)\)
\(\chi_{3320}(963,\cdot)\)
\(\chi_{3320}(987,\cdot)\)
\(\chi_{3320}(1043,\cdot)\)
\(\chi_{3320}(1067,\cdot)\)
\(\chi_{3320}(1267,\cdot)\)
\(\chi_{3320}(1307,\cdot)\)
\(\chi_{3320}(1347,\cdot)\)
...
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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 };
\((831,1661,2657,1081)\) → \((-1,-1,i,e\left(\frac{73}{82}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 3320 }(587, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{139}{164}\right)\) | \(e\left(\frac{143}{164}\right)\) | \(e\left(\frac{57}{82}\right)\) | \(e\left(\frac{15}{41}\right)\) | \(e\left(\frac{131}{164}\right)\) | \(e\left(\frac{17}{164}\right)\) | \(e\left(\frac{14}{41}\right)\) | \(e\left(\frac{59}{82}\right)\) | \(e\left(\frac{109}{164}\right)\) | \(e\left(\frac{89}{164}\right)\) |
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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)
