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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(920, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,22,11,18]))
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gp:[g,chi] = znchar(Mod(517, 920))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("920.517");
| Modulus: | \(920\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(920\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(44\) |
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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;
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| 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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\(\chi_{920}(37,\cdot)\)
\(\chi_{920}(53,\cdot)\)
\(\chi_{920}(157,\cdot)\)
\(\chi_{920}(237,\cdot)\)
\(\chi_{920}(293,\cdot)\)
\(\chi_{920}(333,\cdot)\)
\(\chi_{920}(373,\cdot)\)
\(\chi_{920}(477,\cdot)\)
\(\chi_{920}(493,\cdot)\)
\(\chi_{920}(517,\cdot)\)
\(\chi_{920}(557,\cdot)\)
\(\chi_{920}(573,\cdot)\)
\(\chi_{920}(613,\cdot)\)
\(\chi_{920}(677,\cdot)\)
\(\chi_{920}(733,\cdot)\)
\(\chi_{920}(757,\cdot)\)
\(\chi_{920}(773,\cdot)\)
\(\chi_{920}(797,\cdot)\)
\(\chi_{920}(893,\cdot)\)
\(\chi_{920}(917,\cdot)\)
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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 };
\((231,461,737,281)\) → \((1,-1,i,e\left(\frac{9}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 920 }(517, a) \) |
\(1\) | \(1\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{4}{11}\right)\) |
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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)
