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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(847, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([220,72]))
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gp:[g,chi] = znchar(Mod(445, 847))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("847.445");
| Modulus: | \(847\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| 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: | \(165\) |
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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);
|
\(\chi_{847}(4,\cdot)\)
\(\chi_{847}(16,\cdot)\)
\(\chi_{847}(25,\cdot)\)
\(\chi_{847}(37,\cdot)\)
\(\chi_{847}(53,\cdot)\)
\(\chi_{847}(58,\cdot)\)
\(\chi_{847}(60,\cdot)\)
\(\chi_{847}(86,\cdot)\)
\(\chi_{847}(93,\cdot)\)
\(\chi_{847}(102,\cdot)\)
\(\chi_{847}(114,\cdot)\)
\(\chi_{847}(135,\cdot)\)
\(\chi_{847}(137,\cdot)\)
\(\chi_{847}(158,\cdot)\)
\(\chi_{847}(163,\cdot)\)
\(\chi_{847}(170,\cdot)\)
\(\chi_{847}(179,\cdot)\)
\(\chi_{847}(191,\cdot)\)
\(\chi_{847}(207,\cdot)\)
\(\chi_{847}(212,\cdot)\)
\(\chi_{847}(214,\cdot)\)
\(\chi_{847}(235,\cdot)\)
\(\chi_{847}(240,\cdot)\)
\(\chi_{847}(247,\cdot)\)
\(\chi_{847}(256,\cdot)\)
\(\chi_{847}(268,\cdot)\)
\(\chi_{847}(284,\cdot)\)
\(\chi_{847}(289,\cdot)\)
\(\chi_{847}(291,\cdot)\)
\(\chi_{847}(312,\cdot)\)
...
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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 };
| Field of values: |
$\Q(\zeta_{165})$ |
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sage:CyclotomicField(chi.multiplicative_order())
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gp:nfinit(polcyclo(charorder(g,chi)))
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magma:CyclotomicField(Order(chi));
|
| Fixed field: |
Number field defined by a degree 165 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((122,365)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{12}{55}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 847 }(445, a) \) |
\(1\) | \(1\) | \(e\left(\frac{91}{165}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{17}{165}\right)\) | \(e\left(\frac{79}{165}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{2}{55}\right)\) |
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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)
