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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(31753, base_ring=CyclotomicField(140))
M = H._module
chi = DirichletCharacter(H, M([45,61]))
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gp:[g,chi] = znchar(Mod(13907, 31753))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("31753.13907");
| Modulus: | \(31753\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(31753\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(140\) |
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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);
|
\(\chi_{31753}(347,\cdot)\)
\(\chi_{31753}(1580,\cdot)\)
\(\chi_{31753}(1681,\cdot)\)
\(\chi_{31753}(1755,\cdot)\)
\(\chi_{31753}(2793,\cdot)\)
\(\chi_{31753}(3019,\cdot)\)
\(\chi_{31753}(3673,\cdot)\)
\(\chi_{31753}(3856,\cdot)\)
\(\chi_{31753}(4857,\cdot)\)
\(\chi_{31753}(5025,\cdot)\)
\(\chi_{31753}(5765,\cdot)\)
\(\chi_{31753}(5844,\cdot)\)
\(\chi_{31753}(5932,\cdot)\)
\(\chi_{31753}(6314,\cdot)\)
\(\chi_{31753}(7573,\cdot)\)
\(\chi_{31753}(8037,\cdot)\)
\(\chi_{31753}(8309,\cdot)\)
\(\chi_{31753}(9048,\cdot)\)
\(\chi_{31753}(10792,\cdot)\)
\(\chi_{31753}(12935,\cdot)\)
\(\chi_{31753}(13907,\cdot)\)
\(\chi_{31753}(14206,\cdot)\)
\(\chi_{31753}(15311,\cdot)\)
\(\chi_{31753}(15473,\cdot)\)
\(\chi_{31753}(16280,\cdot)\)
\(\chi_{31753}(16442,\cdot)\)
\(\chi_{31753}(17547,\cdot)\)
\(\chi_{31753}(17846,\cdot)\)
\(\chi_{31753}(18818,\cdot)\)
\(\chi_{31753}(20961,\cdot)\)
...
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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 };
\((20795,10962)\) → \((e\left(\frac{9}{28}\right),e\left(\frac{61}{140}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 31753 }(13907, a) \) |
\(1\) | \(1\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{53}{70}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{101}{140}\right)\) | \(-1\) | \(e\left(\frac{61}{70}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{3}{140}\right)\) |
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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)
