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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(582, base_ring=CyclotomicField(96))
M = H._module
chi = DirichletCharacter(H, M([48,55]))
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gp:[g,chi] = znchar(Mod(251, 582))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("582.251");
| Modulus: | \(582\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(291\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(96\) |
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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 \(\chi_{291}(251,\cdot)\) |
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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);
|
\(\chi_{582}(5,\cdot)\)
\(\chi_{582}(17,\cdot)\)
\(\chi_{582}(23,\cdot)\)
\(\chi_{582}(29,\cdot)\)
\(\chi_{582}(41,\cdot)\)
\(\chi_{582}(59,\cdot)\)
\(\chi_{582}(71,\cdot)\)
\(\chi_{582}(83,\cdot)\)
\(\chi_{582}(107,\cdot)\)
\(\chi_{582}(137,\cdot)\)
\(\chi_{582}(155,\cdot)\)
\(\chi_{582}(173,\cdot)\)
\(\chi_{582}(179,\cdot)\)
\(\chi_{582}(209,\cdot)\)
\(\chi_{582}(215,\cdot)\)
\(\chi_{582}(233,\cdot)\)
\(\chi_{582}(251,\cdot)\)
\(\chi_{582}(281,\cdot)\)
\(\chi_{582}(305,\cdot)\)
\(\chi_{582}(317,\cdot)\)
\(\chi_{582}(329,\cdot)\)
\(\chi_{582}(347,\cdot)\)
\(\chi_{582}(359,\cdot)\)
\(\chi_{582}(365,\cdot)\)
\(\chi_{582}(371,\cdot)\)
\(\chi_{582}(383,\cdot)\)
\(\chi_{582}(395,\cdot)\)
\(\chi_{582}(401,\cdot)\)
\(\chi_{582}(425,\cdot)\)
\(\chi_{582}(545,\cdot)\)
...
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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 };
\((389,199)\) → \((-1,e\left(\frac{55}{96}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 582 }(251, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{96}\right)\) | \(e\left(\frac{73}{96}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{31}{96}\right)\) | \(e\left(\frac{47}{96}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{59}{96}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{91}{96}\right)\) | \(e\left(\frac{17}{48}\right)\) |
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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)
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sage:chi.gauss_sum(a)
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gp:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
