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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5767, base_ring=CyclotomicField(468))
M = H._module
chi = DirichletCharacter(H, M([13,204]))
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gp:[g,chi] = znchar(Mod(171, 5767))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5767.171");
| Modulus: | \(5767\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5767\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(468\) |
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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);
|
\(\chi_{5767}(25,\cdot)\)
\(\chi_{5767}(50,\cdot)\)
\(\chi_{5767}(92,\cdot)\)
\(\chi_{5767}(111,\cdot)\)
\(\chi_{5767}(121,\cdot)\)
\(\chi_{5767}(169,\cdot)\)
\(\chi_{5767}(171,\cdot)\)
\(\chi_{5767}(184,\cdot)\)
\(\chi_{5767}(194,\cdot)\)
\(\chi_{5767}(200,\cdot)\)
\(\chi_{5767}(231,\cdot)\)
\(\chi_{5767}(242,\cdot)\)
\(\chi_{5767}(269,\cdot)\)
\(\chi_{5767}(273,\cdot)\)
\(\chi_{5767}(286,\cdot)\)
\(\chi_{5767}(327,\cdot)\)
\(\chi_{5767}(342,\cdot)\)
\(\chi_{5767}(388,\cdot)\)
\(\chi_{5767}(400,\cdot)\)
\(\chi_{5767}(426,\cdot)\)
\(\chi_{5767}(444,\cdot)\)
\(\chi_{5767}(546,\cdot)\)
\(\chi_{5767}(572,\cdot)\)
\(\chi_{5767}(578,\cdot)\)
\(\chi_{5767}(676,\cdot)\)
\(\chi_{5767}(724,\cdot)\)
\(\chi_{5767}(736,\cdot)\)
\(\chi_{5767}(742,\cdot)\)
\(\chi_{5767}(755,\cdot)\)
\(\chi_{5767}(882,\cdot)\)
...
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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 };
\((1976,1899)\) → \((e\left(\frac{1}{36}\right),e\left(\frac{17}{39}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 5767 }(171, a) \) |
\(1\) | \(1\) | \(e\left(\frac{113}{117}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{109}{117}\right)\) | \(e\left(\frac{25}{468}\right)\) | \(e\left(\frac{133}{234}\right)\) | \(e\left(\frac{1}{52}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{1}{52}\right)\) | \(e\left(\frac{79}{468}\right)\) |
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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)
