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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1491, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([105,35,81]))
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gp:[g,chi] = znchar(Mod(731, 1491))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1491.731");
| Modulus: | \(1491\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1491\) |
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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: | odd |
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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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\(\chi_{1491}(47,\cdot)\)
\(\chi_{1491}(59,\cdot)\)
\(\chi_{1491}(68,\cdot)\)
\(\chi_{1491}(164,\cdot)\)
\(\chi_{1491}(173,\cdot)\)
\(\chi_{1491}(194,\cdot)\)
\(\chi_{1491}(248,\cdot)\)
\(\chi_{1491}(257,\cdot)\)
\(\chi_{1491}(269,\cdot)\)
\(\chi_{1491}(278,\cdot)\)
\(\chi_{1491}(353,\cdot)\)
\(\chi_{1491}(362,\cdot)\)
\(\chi_{1491}(383,\cdot)\)
\(\chi_{1491}(416,\cdot)\)
\(\chi_{1491}(437,\cdot)\)
\(\chi_{1491}(479,\cdot)\)
\(\chi_{1491}(488,\cdot)\)
\(\chi_{1491}(530,\cdot)\)
\(\chi_{1491}(635,\cdot)\)
\(\chi_{1491}(698,\cdot)\)
\(\chi_{1491}(731,\cdot)\)
\(\chi_{1491}(752,\cdot)\)
\(\chi_{1491}(773,\cdot)\)
\(\chi_{1491}(794,\cdot)\)
\(\chi_{1491}(803,\cdot)\)
\(\chi_{1491}(836,\cdot)\)
\(\chi_{1491}(887,\cdot)\)
\(\chi_{1491}(899,\cdot)\)
\(\chi_{1491}(908,\cdot)\)
\(\chi_{1491}(920,\cdot)\)
...
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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 };
\((995,640,1072)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{27}{70}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1491 }(731, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{31}{210}\right)\) | \(e\left(\frac{31}{105}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{31}{70}\right)\) | \(e\left(\frac{59}{210}\right)\) | \(e\left(\frac{13}{105}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{62}{105}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{210}\right)\) |
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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)
