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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2365, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([105,147,115]))
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gp:[g,chi] = znchar(Mod(249, 2365))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2365.249");
| Modulus: | \(2365\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2365\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(210\) |
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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);
|
\(\chi_{2365}(19,\cdot)\)
\(\chi_{2365}(29,\cdot)\)
\(\chi_{2365}(134,\cdot)\)
\(\chi_{2365}(149,\cdot)\)
\(\chi_{2365}(184,\cdot)\)
\(\chi_{2365}(244,\cdot)\)
\(\chi_{2365}(249,\cdot)\)
\(\chi_{2365}(304,\cdot)\)
\(\chi_{2365}(349,\cdot)\)
\(\chi_{2365}(459,\cdot)\)
\(\chi_{2365}(464,\cdot)\)
\(\chi_{2365}(519,\cdot)\)
\(\chi_{2365}(534,\cdot)\)
\(\chi_{2365}(579,\cdot)\)
\(\chi_{2365}(589,\cdot)\)
\(\chi_{2365}(679,\cdot)\)
\(\chi_{2365}(734,\cdot)\)
\(\chi_{2365}(794,\cdot)\)
\(\chi_{2365}(964,\cdot)\)
\(\chi_{2365}(974,\cdot)\)
\(\chi_{2365}(1009,\cdot)\)
\(\chi_{2365}(1019,\cdot)\)
\(\chi_{2365}(1179,\cdot)\)
\(\chi_{2365}(1194,\cdot)\)
\(\chi_{2365}(1234,\cdot)\)
\(\chi_{2365}(1359,\cdot)\)
\(\chi_{2365}(1394,\cdot)\)
\(\chi_{2365}(1404,\cdot)\)
\(\chi_{2365}(1449,\cdot)\)
\(\chi_{2365}(1524,\cdot)\)
...
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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 };
\((947,431,1981)\) → \((-1,e\left(\frac{7}{10}\right),e\left(\frac{23}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 2365 }(249, a) \) |
\(1\) | \(1\) | \(e\left(\frac{69}{70}\right)\) | \(e\left(\frac{68}{105}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{67}{70}\right)\) | \(e\left(\frac{31}{105}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{76}{105}\right)\) | \(e\left(\frac{58}{105}\right)\) |
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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)
