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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(721, base_ring=CyclotomicField(102))
M = H._module
chi = DirichletCharacter(H, M([17,1]))
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gp:[g,chi] = znchar(Mod(108, 721))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("721.108");
| Modulus: | \(721\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(721\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(102\) |
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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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\(\chi_{721}(45,\cdot)\)
\(\chi_{721}(54,\cdot)\)
\(\chi_{721}(75,\cdot)\)
\(\chi_{721}(108,\cdot)\)
\(\chi_{721}(115,\cdot)\)
\(\chi_{721}(138,\cdot)\)
\(\chi_{721}(143,\cdot)\)
\(\chi_{721}(180,\cdot)\)
\(\chi_{721}(187,\cdot)\)
\(\chi_{721}(199,\cdot)\)
\(\chi_{721}(227,\cdot)\)
\(\chi_{721}(250,\cdot)\)
\(\chi_{721}(276,\cdot)\)
\(\chi_{721}(360,\cdot)\)
\(\chi_{721}(374,\cdot)\)
\(\chi_{721}(383,\cdot)\)
\(\chi_{721}(395,\cdot)\)
\(\chi_{721}(418,\cdot)\)
\(\chi_{721}(423,\cdot)\)
\(\chi_{721}(432,\cdot)\)
\(\chi_{721}(460,\cdot)\)
\(\chi_{721}(465,\cdot)\)
\(\chi_{721}(474,\cdot)\)
\(\chi_{721}(479,\cdot)\)
\(\chi_{721}(500,\cdot)\)
\(\chi_{721}(558,\cdot)\)
\(\chi_{721}(593,\cdot)\)
\(\chi_{721}(600,\cdot)\)
\(\chi_{721}(689,\cdot)\)
\(\chi_{721}(705,\cdot)\)
...
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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 };
\((619,211)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{1}{102}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 721 }(108, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{28}{51}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{43}{51}\right)\) | \(e\left(\frac{16}{51}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{5}{51}\right)\) | \(e\left(\frac{31}{51}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{4}{51}\right)\) |
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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)
