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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(695, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([23,12]))
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gp:[g,chi] = znchar(Mod(239, 695))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("695.239");
| Modulus: | \(695\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(695\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(46\) |
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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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\(\chi_{695}(34,\cdot)\)
\(\chi_{695}(44,\cdot)\)
\(\chi_{695}(64,\cdot)\)
\(\chi_{695}(79,\cdot)\)
\(\chi_{695}(129,\cdot)\)
\(\chi_{695}(184,\cdot)\)
\(\chi_{695}(194,\cdot)\)
\(\chi_{695}(204,\cdot)\)
\(\chi_{695}(219,\cdot)\)
\(\chi_{695}(239,\cdot)\)
\(\chi_{695}(264,\cdot)\)
\(\chi_{695}(284,\cdot)\)
\(\chi_{695}(314,\cdot)\)
\(\chi_{695}(369,\cdot)\)
\(\chi_{695}(384,\cdot)\)
\(\chi_{695}(394,\cdot)\)
\(\chi_{695}(409,\cdot)\)
\(\chi_{695}(469,\cdot)\)
\(\chi_{695}(474,\cdot)\)
\(\chi_{695}(494,\cdot)\)
\(\chi_{695}(529,\cdot)\)
\(\chi_{695}(619,\cdot)\)
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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 };
\((557,141)\) → \((-1,e\left(\frac{6}{23}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 695 }(239, a) \) |
\(1\) | \(1\) | \(e\left(\frac{35}{46}\right)\) | \(e\left(\frac{9}{46}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{25}{46}\right)\) | \(e\left(\frac{13}{46}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{33}{46}\right)\) | \(e\left(\frac{9}{46}\right)\) |
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sage:chi(x) # x integer
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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)
