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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(66755, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([117,83,132]))
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gp:[g,chi] = znchar(Mod(15178, 66755))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("66755.15178");
| Modulus: | \(66755\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(66755\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(156\) |
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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);
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\(\chi_{66755}(97,\cdot)\)
\(\chi_{66755}(1918,\cdot)\)
\(\chi_{66755}(2827,\cdot)\)
\(\chi_{66755}(3607,\cdot)\)
\(\chi_{66755}(3933,\cdot)\)
\(\chi_{66755}(4647,\cdot)\)
\(\chi_{66755}(7572,\cdot)\)
\(\chi_{66755}(8123,\cdot)\)
\(\chi_{66755}(8578,\cdot)\)
\(\chi_{66755}(14983,\cdot)\)
\(\chi_{66755}(15178,\cdot)\)
\(\chi_{66755}(17548,\cdot)\)
\(\chi_{66755}(18982,\cdot)\)
\(\chi_{66755}(19077,\cdot)\)
\(\chi_{66755}(20087,\cdot)\)
\(\chi_{66755}(27592,\cdot)\)
\(\chi_{66755}(27623,\cdot)\)
\(\chi_{66755}(30877,\cdot)\)
\(\chi_{66755}(31267,\cdot)\)
\(\chi_{66755}(31427,\cdot)\)
\(\chi_{66755}(32013,\cdot)\)
\(\chi_{66755}(35098,\cdot)\)
\(\chi_{66755}(36723,\cdot)\)
\(\chi_{66755}(37247,\cdot)\)
\(\chi_{66755}(37377,\cdot)\)
\(\chi_{66755}(37958,\cdot)\)
\(\chi_{66755}(38807,\cdot)\)
\(\chi_{66755}(39747,\cdot)\)
\(\chi_{66755}(40038,\cdot)\)
\(\chi_{66755}(42248,\cdot)\)
...
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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 };
\((13352,33971,38871)\) → \((-i,e\left(\frac{83}{156}\right),e\left(\frac{11}{13}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 66755 }(15178, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{156}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{115}{156}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(1\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{53}{156}\right)\) | \(e\left(\frac{21}{52}\right)\) | \(e\left(\frac{5}{26}\right)\) |
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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)
