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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8464, base_ring=CyclotomicField(92))
M = H._module
chi = DirichletCharacter(H, M([46,69,90]))
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gp:[g,chi] = znchar(Mod(1747, 8464))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8464.1747");
| 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: | \(92\) |
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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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\(\chi_{8464}(91,\cdot)\)
\(\chi_{8464}(275,\cdot)\)
\(\chi_{8464}(459,\cdot)\)
\(\chi_{8464}(643,\cdot)\)
\(\chi_{8464}(827,\cdot)\)
\(\chi_{8464}(1011,\cdot)\)
\(\chi_{8464}(1195,\cdot)\)
\(\chi_{8464}(1379,\cdot)\)
\(\chi_{8464}(1563,\cdot)\)
\(\chi_{8464}(1747,\cdot)\)
\(\chi_{8464}(1931,\cdot)\)
\(\chi_{8464}(2299,\cdot)\)
\(\chi_{8464}(2483,\cdot)\)
\(\chi_{8464}(2667,\cdot)\)
\(\chi_{8464}(2851,\cdot)\)
\(\chi_{8464}(3035,\cdot)\)
\(\chi_{8464}(3219,\cdot)\)
\(\chi_{8464}(3403,\cdot)\)
\(\chi_{8464}(3587,\cdot)\)
\(\chi_{8464}(3771,\cdot)\)
\(\chi_{8464}(3955,\cdot)\)
\(\chi_{8464}(4139,\cdot)\)
\(\chi_{8464}(4323,\cdot)\)
\(\chi_{8464}(4507,\cdot)\)
\(\chi_{8464}(4691,\cdot)\)
\(\chi_{8464}(4875,\cdot)\)
\(\chi_{8464}(5059,\cdot)\)
\(\chi_{8464}(5243,\cdot)\)
\(\chi_{8464}(5427,\cdot)\)
\(\chi_{8464}(5611,\cdot)\)
...
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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 };
\((7407,2117,6353)\) → \((-1,-i,e\left(\frac{45}{46}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 8464 }(1747, a) \) |
\(1\) | \(1\) | \(e\left(\frac{37}{92}\right)\) | \(e\left(\frac{67}{92}\right)\) | \(e\left(\frac{9}{46}\right)\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{45}{92}\right)\) | \(e\left(\frac{11}{92}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{83}{92}\right)\) | \(e\left(\frac{55}{92}\right)\) |
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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)
