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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3661, base_ring=CyclotomicField(522))
M = H._module
chi = DirichletCharacter(H, M([435,455]))
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gp:[g,chi] = znchar(Mod(236, 3661))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3661.236");
| Modulus: | \(3661\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3661\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(522\) |
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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);
|
\(\chi_{3661}(12,\cdot)\)
\(\chi_{3661}(45,\cdot)\)
\(\chi_{3661}(75,\cdot)\)
\(\chi_{3661}(103,\cdot)\)
\(\chi_{3661}(152,\cdot)\)
\(\chi_{3661}(213,\cdot)\)
\(\chi_{3661}(222,\cdot)\)
\(\chi_{3661}(227,\cdot)\)
\(\chi_{3661}(229,\cdot)\)
\(\chi_{3661}(236,\cdot)\)
\(\chi_{3661}(276,\cdot)\)
\(\chi_{3661}(355,\cdot)\)
\(\chi_{3661}(374,\cdot)\)
\(\chi_{3661}(388,\cdot)\)
\(\chi_{3661}(404,\cdot)\)
\(\chi_{3661}(416,\cdot)\)
\(\chi_{3661}(423,\cdot)\)
\(\chi_{3661}(446,\cdot)\)
\(\chi_{3661}(460,\cdot)\)
\(\chi_{3661}(507,\cdot)\)
\(\chi_{3661}(516,\cdot)\)
\(\chi_{3661}(528,\cdot)\)
\(\chi_{3661}(544,\cdot)\)
\(\chi_{3661}(570,\cdot)\)
\(\chi_{3661}(579,\cdot)\)
\(\chi_{3661}(605,\cdot)\)
\(\chi_{3661}(621,\cdot)\)
\(\chi_{3661}(663,\cdot)\)
\(\chi_{3661}(705,\cdot)\)
\(\chi_{3661}(712,\cdot)\)
...
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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 };
\((3139,2094)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{455}{522}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 3661 }(236, a) \) |
\(1\) | \(1\) | \(e\left(\frac{281}{522}\right)\) | \(e\left(\frac{62}{87}\right)\) | \(e\left(\frac{20}{261}\right)\) | \(e\left(\frac{229}{261}\right)\) | \(e\left(\frac{131}{522}\right)\) | \(e\left(\frac{107}{174}\right)\) | \(e\left(\frac{37}{87}\right)\) | \(e\left(\frac{217}{522}\right)\) | \(e\left(\frac{59}{87}\right)\) | \(e\left(\frac{206}{261}\right)\) |
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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)
