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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8015, base_ring=CyclotomicField(76))
M = H._module
chi = DirichletCharacter(H, M([57,38,47]))
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gp:[g,chi] = znchar(Mod(5158, 8015))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8015.5158");
| Modulus: | \(8015\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(8015\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(76\) |
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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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\(\chi_{8015}(13,\cdot)\)
\(\chi_{8015}(237,\cdot)\)
\(\chi_{8015}(832,\cdot)\)
\(\chi_{8015}(937,\cdot)\)
\(\chi_{8015}(1238,\cdot)\)
\(\chi_{8015}(1497,\cdot)\)
\(\chi_{8015}(1938,\cdot)\)
\(\chi_{8015}(2197,\cdot)\)
\(\chi_{8015}(2498,\cdot)\)
\(\chi_{8015}(2603,\cdot)\)
\(\chi_{8015}(3198,\cdot)\)
\(\chi_{8015}(3422,\cdot)\)
\(\chi_{8015}(3457,\cdot)\)
\(\chi_{8015}(3632,\cdot)\)
\(\chi_{8015}(3807,\cdot)\)
\(\chi_{8015}(3863,\cdot)\)
\(\chi_{8015}(4297,\cdot)\)
\(\chi_{8015}(4353,\cdot)\)
\(\chi_{8015}(4528,\cdot)\)
\(\chi_{8015}(4668,\cdot)\)
\(\chi_{8015}(4843,\cdot)\)
\(\chi_{8015}(5158,\cdot)\)
\(\chi_{8015}(5368,\cdot)\)
\(\chi_{8015}(5382,\cdot)\)
\(\chi_{8015}(6068,\cdot)\)
\(\chi_{8015}(6082,\cdot)\)
\(\chi_{8015}(6292,\cdot)\)
\(\chi_{8015}(6607,\cdot)\)
\(\chi_{8015}(6782,\cdot)\)
\(\chi_{8015}(6922,\cdot)\)
...
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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 };
\((3207,4581,4586)\) → \((-i,-1,e\left(\frac{47}{76}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 8015 }(5158, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{29}{76}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{9}{76}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{65}{76}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{18}{19}\right)\) |
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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)
