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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(13005, base_ring=CyclotomicField(816))
M = H._module
chi = DirichletCharacter(H, M([544,612,159]))
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gp:[g,chi] = znchar(Mod(1348, 13005))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("13005.1348");
| Modulus: | \(13005\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(13005\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(816\) |
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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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\(\chi_{13005}(7,\cdot)\)
\(\chi_{13005}(88,\cdot)\)
\(\chi_{13005}(112,\cdot)\)
\(\chi_{13005}(133,\cdot)\)
\(\chi_{13005}(142,\cdot)\)
\(\chi_{13005}(148,\cdot)\)
\(\chi_{13005}(232,\cdot)\)
\(\chi_{13005}(328,\cdot)\)
\(\chi_{13005}(367,\cdot)\)
\(\chi_{13005}(403,\cdot)\)
\(\chi_{13005}(517,\cdot)\)
\(\chi_{13005}(583,\cdot)\)
\(\chi_{13005}(598,\cdot)\)
\(\chi_{13005}(652,\cdot)\)
\(\chi_{13005}(742,\cdot)\)
\(\chi_{13005}(772,\cdot)\)
\(\chi_{13005}(853,\cdot)\)
\(\chi_{13005}(877,\cdot)\)
\(\chi_{13005}(898,\cdot)\)
\(\chi_{13005}(913,\cdot)\)
\(\chi_{13005}(997,\cdot)\)
\(\chi_{13005}(1093,\cdot)\)
\(\chi_{13005}(1132,\cdot)\)
\(\chi_{13005}(1168,\cdot)\)
\(\chi_{13005}(1282,\cdot)\)
\(\chi_{13005}(1348,\cdot)\)
\(\chi_{13005}(1363,\cdot)\)
\(\chi_{13005}(1408,\cdot)\)
\(\chi_{13005}(1417,\cdot)\)
\(\chi_{13005}(1507,\cdot)\)
...
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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 };
\((2891,2602,2026)\) → \((e\left(\frac{2}{3}\right),-i,e\left(\frac{53}{272}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(19\) | \(22\) |
| \( \chi_{ 13005 }(1348, a) \) |
\(1\) | \(1\) | \(e\left(\frac{179}{408}\right)\) | \(e\left(\frac{179}{204}\right)\) | \(e\left(\frac{505}{816}\right)\) | \(e\left(\frac{43}{136}\right)\) | \(e\left(\frac{121}{816}\right)\) | \(e\left(\frac{79}{102}\right)\) | \(e\left(\frac{47}{816}\right)\) | \(e\left(\frac{77}{102}\right)\) | \(e\left(\frac{31}{136}\right)\) | \(e\left(\frac{479}{816}\right)\) |
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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)
