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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2021, base_ring=CyclotomicField(322))
M = H._module
chi = DirichletCharacter(H, M([69,56]))
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gp:[g,chi] = znchar(Mod(290, 2021))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2021.290");
| Modulus: | \(2021\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2021\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(322\) |
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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);
|
\(\chi_{2021}(2,\cdot)\)
\(\chi_{2021}(8,\cdot)\)
\(\chi_{2021}(27,\cdot)\)
\(\chi_{2021}(32,\cdot)\)
\(\chi_{2021}(51,\cdot)\)
\(\chi_{2021}(65,\cdot)\)
\(\chi_{2021}(75,\cdot)\)
\(\chi_{2021}(108,\cdot)\)
\(\chi_{2021}(118,\cdot)\)
\(\chi_{2021}(131,\cdot)\)
\(\chi_{2021}(168,\cdot)\)
\(\chi_{2021}(194,\cdot)\)
\(\chi_{2021}(204,\cdot)\)
\(\chi_{2021}(237,\cdot)\)
\(\chi_{2021}(242,\cdot)\)
\(\chi_{2021}(247,\cdot)\)
\(\chi_{2021}(260,\cdot)\)
\(\chi_{2021}(285,\cdot)\)
\(\chi_{2021}(290,\cdot)\)
\(\chi_{2021}(303,\cdot)\)
\(\chi_{2021}(309,\cdot)\)
\(\chi_{2021}(333,\cdot)\)
\(\chi_{2021}(346,\cdot)\)
\(\chi_{2021}(366,\cdot)\)
\(\chi_{2021}(371,\cdot)\)
\(\chi_{2021}(383,\cdot)\)
\(\chi_{2021}(426,\cdot)\)
\(\chi_{2021}(432,\cdot)\)
\(\chi_{2021}(457,\cdot)\)
\(\chi_{2021}(495,\cdot)\)
...
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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 };
\((1035,1979)\) → \((e\left(\frac{3}{14}\right),e\left(\frac{4}{23}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 2021 }(290, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{295}{322}\right)\) | \(e\left(\frac{223}{322}\right)\) | \(e\left(\frac{134}{161}\right)\) | \(e\left(\frac{171}{322}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{3}{46}\right)\) | \(e\left(\frac{241}{322}\right)\) | \(e\left(\frac{62}{161}\right)\) | \(e\left(\frac{72}{161}\right)\) | \(e\left(\frac{104}{161}\right)\) |
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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)
