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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(12095, base_ring=CyclotomicField(116))
M = H._module
chi = DirichletCharacter(H, M([87,29,22]))
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gp:[g,chi] = znchar(Mod(278, 12095))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("12095.278");
| Modulus: | \(12095\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(12095\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(116\) |
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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);
|
\(\chi_{12095}(73,\cdot)\)
\(\chi_{12095}(132,\cdot)\)
\(\chi_{12095}(278,\cdot)\)
\(\chi_{12095}(337,\cdot)\)
\(\chi_{12095}(483,\cdot)\)
\(\chi_{12095}(542,\cdot)\)
\(\chi_{12095}(688,\cdot)\)
\(\chi_{12095}(747,\cdot)\)
\(\chi_{12095}(893,\cdot)\)
\(\chi_{12095}(952,\cdot)\)
\(\chi_{12095}(1508,\cdot)\)
\(\chi_{12095}(1567,\cdot)\)
\(\chi_{12095}(1713,\cdot)\)
\(\chi_{12095}(1772,\cdot)\)
\(\chi_{12095}(1918,\cdot)\)
\(\chi_{12095}(1977,\cdot)\)
\(\chi_{12095}(2533,\cdot)\)
\(\chi_{12095}(2592,\cdot)\)
\(\chi_{12095}(2738,\cdot)\)
\(\chi_{12095}(2797,\cdot)\)
\(\chi_{12095}(2943,\cdot)\)
\(\chi_{12095}(3002,\cdot)\)
\(\chi_{12095}(3558,\cdot)\)
\(\chi_{12095}(3617,\cdot)\)
\(\chi_{12095}(4173,\cdot)\)
\(\chi_{12095}(4232,\cdot)\)
\(\chi_{12095}(4583,\cdot)\)
\(\chi_{12095}(4642,\cdot)\)
\(\chi_{12095}(4993,\cdot)\)
\(\chi_{12095}(5052,\cdot)\)
...
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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 };
\((9677,4721,3896)\) → \((-i,i,e\left(\frac{11}{58}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 12095 }(278, a) \) |
\(1\) | \(1\) | \(e\left(\frac{51}{116}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{51}{58}\right)\) | \(e\left(\frac{107}{116}\right)\) | \(e\left(\frac{53}{58}\right)\) | \(e\left(\frac{37}{116}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{57}{116}\right)\) | \(e\left(\frac{21}{58}\right)\) | \(e\left(\frac{31}{58}\right)\) |
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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)
