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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6258, base_ring=CyclotomicField(444))
M = H._module
chi = DirichletCharacter(H, M([222,148,129]))
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gp:[g,chi] = znchar(Mod(611, 6258))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6258.611");
| Modulus: | \(6258\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3129\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(444\) |
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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: | no, induced from \(\chi_{3129}(611,\cdot)\) |
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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);
|
\(\chi_{6258}(11,\cdot)\)
\(\chi_{6258}(23,\cdot)\)
\(\chi_{6258}(65,\cdot)\)
\(\chi_{6258}(137,\cdot)\)
\(\chi_{6258}(221,\cdot)\)
\(\chi_{6258}(233,\cdot)\)
\(\chi_{6258}(275,\cdot)\)
\(\chi_{6258}(389,\cdot)\)
\(\chi_{6258}(485,\cdot)\)
\(\chi_{6258}(569,\cdot)\)
\(\chi_{6258}(599,\cdot)\)
\(\chi_{6258}(611,\cdot)\)
\(\chi_{6258}(653,\cdot)\)
\(\chi_{6258}(683,\cdot)\)
\(\chi_{6258}(695,\cdot)\)
\(\chi_{6258}(737,\cdot)\)
\(\chi_{6258}(779,\cdot)\)
\(\chi_{6258}(851,\cdot)\)
\(\chi_{6258}(905,\cdot)\)
\(\chi_{6258}(935,\cdot)\)
\(\chi_{6258}(977,\cdot)\)
\(\chi_{6258}(1031,\cdot)\)
\(\chi_{6258}(1061,\cdot)\)
\(\chi_{6258}(1103,\cdot)\)
\(\chi_{6258}(1115,\cdot)\)
\(\chi_{6258}(1271,\cdot)\)
\(\chi_{6258}(1283,\cdot)\)
\(\chi_{6258}(1355,\cdot)\)
\(\chi_{6258}(1397,\cdot)\)
\(\chi_{6258}(1439,\cdot)\)
...
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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 };
\((2087,5365,4621)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{43}{148}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 6258 }(611, a) \) |
\(1\) | \(1\) | \(e\left(\frac{85}{222}\right)\) | \(e\left(\frac{223}{444}\right)\) | \(e\left(\frac{59}{148}\right)\) | \(e\left(\frac{191}{222}\right)\) | \(e\left(\frac{8}{111}\right)\) | \(e\left(\frac{341}{444}\right)\) | \(e\left(\frac{85}{111}\right)\) | \(e\left(\frac{27}{74}\right)\) | \(e\left(\frac{76}{111}\right)\) | \(e\left(\frac{65}{111}\right)\) |
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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)
