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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3445, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([117,65,129]))
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gp:[g,chi] = znchar(Mod(1163, 3445))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3445.1163");
| Modulus: | \(3445\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3445\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(156\) |
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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);
|
\(\chi_{3445}(58,\cdot)\)
\(\chi_{3445}(72,\cdot)\)
\(\chi_{3445}(137,\cdot)\)
\(\chi_{3445}(232,\cdot)\)
\(\chi_{3445}(267,\cdot)\)
\(\chi_{3445}(297,\cdot)\)
\(\chi_{3445}(383,\cdot)\)
\(\chi_{3445}(397,\cdot)\)
\(\chi_{3445}(427,\cdot)\)
\(\chi_{3445}(548,\cdot)\)
\(\chi_{3445}(622,\cdot)\)
\(\chi_{3445}(787,\cdot)\)
\(\chi_{3445}(968,\cdot)\)
\(\chi_{3445}(1068,\cdot)\)
\(\chi_{3445}(1163,\cdot)\)
\(\chi_{3445}(1207,\cdot)\)
\(\chi_{3445}(1293,\cdot)\)
\(\chi_{3445}(1307,\cdot)\)
\(\chi_{3445}(1358,\cdot)\)
\(\chi_{3445}(1458,\cdot)\)
\(\chi_{3445}(1532,\cdot)\)
\(\chi_{3445}(1588,\cdot)\)
\(\chi_{3445}(1662,\cdot)\)
\(\chi_{3445}(1718,\cdot)\)
\(\chi_{3445}(1727,\cdot)\)
\(\chi_{3445}(1783,\cdot)\)
\(\chi_{3445}(1857,\cdot)\)
\(\chi_{3445}(1913,\cdot)\)
\(\chi_{3445}(1987,\cdot)\)
\(\chi_{3445}(2087,\cdot)\)
...
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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 };
\((2757,2121,2016)\) → \((-i,e\left(\frac{5}{12}\right),e\left(\frac{43}{52}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 3445 }(1163, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{155}{156}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{151}{156}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{51}{52}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{137}{156}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{47}{52}\right)\) |
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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)
