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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(31753, base_ring=CyclotomicField(280))
M = H._module
chi = DirichletCharacter(H, M([255,107]))
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gp:[g,chi] = znchar(Mod(7067, 31753))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("31753.7067");
| 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: | \(280\) |
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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);
|
\(\chi_{31753}(176,\cdot)\)
\(\chi_{31753}(251,\cdot)\)
\(\chi_{31753}(257,\cdot)\)
\(\chi_{31753}(817,\cdot)\)
\(\chi_{31753}(1053,\cdot)\)
\(\chi_{31753}(1121,\cdot)\)
\(\chi_{31753}(1218,\cdot)\)
\(\chi_{31753}(1480,\cdot)\)
\(\chi_{31753}(1569,\cdot)\)
\(\chi_{31753}(2021,\cdot)\)
\(\chi_{31753}(2508,\cdot)\)
\(\chi_{31753}(2573,\cdot)\)
\(\chi_{31753}(2907,\cdot)\)
\(\chi_{31753}(3015,\cdot)\)
\(\chi_{31753}(3704,\cdot)\)
\(\chi_{31753}(3986,\cdot)\)
\(\chi_{31753}(4469,\cdot)\)
\(\chi_{31753}(4509,\cdot)\)
\(\chi_{31753}(4583,\cdot)\)
\(\chi_{31753}(4655,\cdot)\)
\(\chi_{31753}(4695,\cdot)\)
\(\chi_{31753}(5449,\cdot)\)
\(\chi_{31753}(6204,\cdot)\)
\(\chi_{31753}(6379,\cdot)\)
\(\chi_{31753}(6852,\cdot)\)
\(\chi_{31753}(7067,\cdot)\)
\(\chi_{31753}(7191,\cdot)\)
\(\chi_{31753}(7397,\cdot)\)
\(\chi_{31753}(7761,\cdot)\)
\(\chi_{31753}(9140,\cdot)\)
...
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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 };
\((20795,10962)\) → \((e\left(\frac{51}{56}\right),e\left(\frac{107}{280}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 31753 }(7067, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{41}{140}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{187}{280}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{117}{140}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{41}{70}\right)\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{3}{40}\right)\) |
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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)
