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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8464, base_ring=CyclotomicField(1012))
M = H._module
chi = DirichletCharacter(H, M([0,253,864]))
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gp:[g,chi] = znchar(Mod(565, 8464))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8464.565");
| Modulus: | \(8464\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(8464\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1012\) |
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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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\(\chi_{8464}(13,\cdot)\)
\(\chi_{8464}(29,\cdot)\)
\(\chi_{8464}(77,\cdot)\)
\(\chi_{8464}(85,\cdot)\)
\(\chi_{8464}(101,\cdot)\)
\(\chi_{8464}(117,\cdot)\)
\(\chi_{8464}(133,\cdot)\)
\(\chi_{8464}(141,\cdot)\)
\(\chi_{8464}(165,\cdot)\)
\(\chi_{8464}(173,\cdot)\)
\(\chi_{8464}(197,\cdot)\)
\(\chi_{8464}(213,\cdot)\)
\(\chi_{8464}(261,\cdot)\)
\(\chi_{8464}(269,\cdot)\)
\(\chi_{8464}(285,\cdot)\)
\(\chi_{8464}(301,\cdot)\)
\(\chi_{8464}(317,\cdot)\)
\(\chi_{8464}(325,\cdot)\)
\(\chi_{8464}(349,\cdot)\)
\(\chi_{8464}(357,\cdot)\)
\(\chi_{8464}(381,\cdot)\)
\(\chi_{8464}(397,\cdot)\)
\(\chi_{8464}(445,\cdot)\)
\(\chi_{8464}(453,\cdot)\)
\(\chi_{8464}(469,\cdot)\)
\(\chi_{8464}(485,\cdot)\)
\(\chi_{8464}(509,\cdot)\)
\(\chi_{8464}(533,\cdot)\)
\(\chi_{8464}(541,\cdot)\)
\(\chi_{8464}(565,\cdot)\)
...
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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 };
\((7407,2117,6353)\) → \((1,i,e\left(\frac{216}{253}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 8464 }(565, a) \) |
\(1\) | \(1\) | \(e\left(\frac{415}{1012}\right)\) | \(e\left(\frac{105}{1012}\right)\) | \(e\left(\frac{321}{506}\right)\) | \(e\left(\frac{415}{506}\right)\) | \(e\left(\frac{593}{1012}\right)\) | \(e\left(\frac{975}{1012}\right)\) | \(e\left(\frac{130}{253}\right)\) | \(e\left(\frac{192}{253}\right)\) | \(e\left(\frac{783}{1012}\right)\) | \(e\left(\frac{45}{1012}\right)\) |
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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)
