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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6664, base_ring=CyclotomicField(168))
M = H._module
chi = DirichletCharacter(H, M([84,0,148,63]))
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gp:[g,chi] = znchar(Mod(2327, 6664))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6664.2327");
| Modulus: | \(6664\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3332\) |
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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 \(\chi_{3332}(2327,\cdot)\) |
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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: | no |
| 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);
|
\(\chi_{6664}(87,\cdot)\)
\(\chi_{6664}(383,\cdot)\)
\(\chi_{6664}(495,\cdot)\)
\(\chi_{6664}(535,\cdot)\)
\(\chi_{6664}(831,\cdot)\)
\(\chi_{6664}(927,\cdot)\)
\(\chi_{6664}(943,\cdot)\)
\(\chi_{6664}(1039,\cdot)\)
\(\chi_{6664}(1335,\cdot)\)
\(\chi_{6664}(1375,\cdot)\)
\(\chi_{6664}(1447,\cdot)\)
\(\chi_{6664}(1487,\cdot)\)
\(\chi_{6664}(1879,\cdot)\)
\(\chi_{6664}(1895,\cdot)\)
\(\chi_{6664}(2287,\cdot)\)
\(\chi_{6664}(2327,\cdot)\)
\(\chi_{6664}(2399,\cdot)\)
\(\chi_{6664}(2439,\cdot)\)
\(\chi_{6664}(2735,\cdot)\)
\(\chi_{6664}(2831,\cdot)\)
\(\chi_{6664}(2847,\cdot)\)
\(\chi_{6664}(2943,\cdot)\)
\(\chi_{6664}(3239,\cdot)\)
\(\chi_{6664}(3279,\cdot)\)
\(\chi_{6664}(3391,\cdot)\)
\(\chi_{6664}(3687,\cdot)\)
\(\chi_{6664}(3783,\cdot)\)
\(\chi_{6664}(3799,\cdot)\)
\(\chi_{6664}(3895,\cdot)\)
\(\chi_{6664}(4191,\cdot)\)
...
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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 };
\((4999,3333,4217,785)\) → \((-1,1,e\left(\frac{37}{42}\right),e\left(\frac{3}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 6664 }(2327, a) \) |
\(1\) | \(1\) | \(e\left(\frac{127}{168}\right)\) | \(e\left(\frac{71}{168}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{61}{168}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{101}{168}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{15}{56}\right)\) |
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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)
