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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(847, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([11,57]))
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gp:[g,chi] = znchar(Mod(626, 847))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("847.626");
| Modulus: | \(847\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
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| Conductor: | \(847\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(66\) |
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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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\(\chi_{847}(10,\cdot)\)
\(\chi_{847}(54,\cdot)\)
\(\chi_{847}(87,\cdot)\)
\(\chi_{847}(131,\cdot)\)
\(\chi_{847}(164,\cdot)\)
\(\chi_{847}(208,\cdot)\)
\(\chi_{847}(285,\cdot)\)
\(\chi_{847}(318,\cdot)\)
\(\chi_{847}(395,\cdot)\)
\(\chi_{847}(439,\cdot)\)
\(\chi_{847}(472,\cdot)\)
\(\chi_{847}(516,\cdot)\)
\(\chi_{847}(549,\cdot)\)
\(\chi_{847}(593,\cdot)\)
\(\chi_{847}(626,\cdot)\)
\(\chi_{847}(670,\cdot)\)
\(\chi_{847}(703,\cdot)\)
\(\chi_{847}(747,\cdot)\)
\(\chi_{847}(780,\cdot)\)
\(\chi_{847}(824,\cdot)\)
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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 };
\((122,365)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{19}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 847 }(626, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{8}{11}\right)\) |
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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)
