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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(76585, base_ring=CyclotomicField(272))
M = H._module
chi = DirichletCharacter(H, M([68,133,136]))
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gp:[g,chi] = znchar(Mod(21782, 76585))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("76585.21782");
| Modulus: | \(76585\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(76585\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(272\) |
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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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\(\chi_{76585}(847,\cdot)\)
\(\chi_{76585}(1218,\cdot)\)
\(\chi_{76585}(1907,\cdot)\)
\(\chi_{76585}(2013,\cdot)\)
\(\chi_{76585}(2543,\cdot)\)
\(\chi_{76585}(3338,\cdot)\)
\(\chi_{76585}(3497,\cdot)\)
\(\chi_{76585}(3762,\cdot)\)
\(\chi_{76585}(5352,\cdot)\)
\(\chi_{76585}(5723,\cdot)\)
\(\chi_{76585}(6412,\cdot)\)
\(\chi_{76585}(6518,\cdot)\)
\(\chi_{76585}(7048,\cdot)\)
\(\chi_{76585}(8002,\cdot)\)
\(\chi_{76585}(8267,\cdot)\)
\(\chi_{76585}(9857,\cdot)\)
\(\chi_{76585}(10228,\cdot)\)
\(\chi_{76585}(11023,\cdot)\)
\(\chi_{76585}(11553,\cdot)\)
\(\chi_{76585}(12348,\cdot)\)
\(\chi_{76585}(12507,\cdot)\)
\(\chi_{76585}(12772,\cdot)\)
\(\chi_{76585}(14362,\cdot)\)
\(\chi_{76585}(14733,\cdot)\)
\(\chi_{76585}(15422,\cdot)\)
\(\chi_{76585}(15528,\cdot)\)
\(\chi_{76585}(16058,\cdot)\)
\(\chi_{76585}(16853,\cdot)\)
\(\chi_{76585}(17012,\cdot)\)
\(\chi_{76585}(17277,\cdot)\)
...
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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 };
\((45952,6361,27456)\) → \((i,e\left(\frac{133}{272}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 76585 }(21782, a) \) |
\(1\) | \(1\) | \(e\left(\frac{89}{136}\right)\) | \(e\left(\frac{201}{272}\right)\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{107}{272}\right)\) | \(e\left(\frac{11}{272}\right)\) | \(e\left(\frac{131}{136}\right)\) | \(e\left(\frac{65}{136}\right)\) | \(e\left(\frac{67}{272}\right)\) | \(e\left(\frac{13}{272}\right)\) | \(e\left(\frac{10}{17}\right)\) |
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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)
