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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8381, base_ring=CyclotomicField(476))
M = H._module
chi = DirichletCharacter(H, M([301,85]))
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gp:[g,chi] = znchar(Mod(1279, 8381))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8381.1279");
| Modulus: | \(8381\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(8381\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(476\) |
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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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\(\chi_{8381}(21,\cdot)\)
\(\chi_{8381}(47,\cdot)\)
\(\chi_{8381}(72,\cdot)\)
\(\chi_{8381}(89,\cdot)\)
\(\chi_{8381}(106,\cdot)\)
\(\chi_{8381}(200,\cdot)\)
\(\chi_{8381}(293,\cdot)\)
\(\chi_{8381}(387,\cdot)\)
\(\chi_{8381}(404,\cdot)\)
\(\chi_{8381}(421,\cdot)\)
\(\chi_{8381}(446,\cdot)\)
\(\chi_{8381}(472,\cdot)\)
\(\chi_{8381}(514,\cdot)\)
\(\chi_{8381}(565,\cdot)\)
\(\chi_{8381}(582,\cdot)\)
\(\chi_{8381}(599,\cdot)\)
\(\chi_{8381}(693,\cdot)\)
\(\chi_{8381}(786,\cdot)\)
\(\chi_{8381}(880,\cdot)\)
\(\chi_{8381}(897,\cdot)\)
\(\chi_{8381}(914,\cdot)\)
\(\chi_{8381}(939,\cdot)\)
\(\chi_{8381}(965,\cdot)\)
\(\chi_{8381}(1007,\cdot)\)
\(\chi_{8381}(1033,\cdot)\)
\(\chi_{8381}(1058,\cdot)\)
\(\chi_{8381}(1075,\cdot)\)
\(\chi_{8381}(1092,\cdot)\)
\(\chi_{8381}(1186,\cdot)\)
\(\chi_{8381}(1279,\cdot)\)
...
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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 };
\((581,8093)\) → \((e\left(\frac{43}{68}\right),e\left(\frac{5}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 8381 }(1279, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{155}{476}\right)\) | \(e\left(\frac{125}{238}\right)\) | \(e\left(\frac{155}{238}\right)\) | \(e\left(\frac{351}{476}\right)\) | \(e\left(\frac{405}{476}\right)\) | \(e\left(\frac{75}{476}\right)\) | \(e\left(\frac{465}{476}\right)\) | \(e\left(\frac{6}{119}\right)\) | \(e\left(\frac{15}{238}\right)\) | \(e\left(\frac{1}{119}\right)\) |
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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)
