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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(62475, base_ring=CyclotomicField(168))
M = H._module
chi = DirichletCharacter(H, M([84,0,124,21]))
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gp:[g,chi] = znchar(Mod(24251, 62475))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("62475.24251");
| Modulus: | \(62475\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2499\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(168\) |
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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 \(\chi_{2499}(1760,\cdot)\) |
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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);
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\(\chi_{62475}(26,\cdot)\)
\(\chi_{62475}(2201,\cdot)\)
\(\chi_{62475}(3776,\cdot)\)
\(\chi_{62475}(4751,\cdot)\)
\(\chi_{62475}(4826,\cdot)\)
\(\chi_{62475}(6326,\cdot)\)
\(\chi_{62475}(7376,\cdot)\)
\(\chi_{62475}(8951,\cdot)\)
\(\chi_{62475}(11126,\cdot)\)
\(\chi_{62475}(12701,\cdot)\)
\(\chi_{62475}(13676,\cdot)\)
\(\chi_{62475}(15251,\cdot)\)
\(\chi_{62475}(15326,\cdot)\)
\(\chi_{62475}(16301,\cdot)\)
\(\chi_{62475}(17876,\cdot)\)
\(\chi_{62475}(20051,\cdot)\)
\(\chi_{62475}(21626,\cdot)\)
\(\chi_{62475}(22601,\cdot)\)
\(\chi_{62475}(22676,\cdot)\)
\(\chi_{62475}(24251,\cdot)\)
\(\chi_{62475}(25226,\cdot)\)
\(\chi_{62475}(26801,\cdot)\)
\(\chi_{62475}(28976,\cdot)\)
\(\chi_{62475}(30551,\cdot)\)
\(\chi_{62475}(31601,\cdot)\)
\(\chi_{62475}(33101,\cdot)\)
\(\chi_{62475}(33176,\cdot)\)
\(\chi_{62475}(34151,\cdot)\)
\(\chi_{62475}(35726,\cdot)\)
\(\chi_{62475}(37901,\cdot)\)
...
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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 };
\((41651,59977,2551,44101)\) → \((-1,1,e\left(\frac{31}{42}\right),e\left(\frac{1}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(19\) | \(22\) | \(23\) | \(26\) |
| \( \chi_{ 62475 }(24251, a) \) |
\(1\) | \(1\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{151}{168}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{71}{168}\right)\) | \(e\left(\frac{25}{84}\right)\) |
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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)
