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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8509, base_ring=CyclotomicField(462))
M = H._module
chi = DirichletCharacter(H, M([119,22]))
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gp:[g,chi] = znchar(Mod(221, 8509))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8509.221");
| Modulus: | \(8509\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(8509\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(462\) |
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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);
|
\(\chi_{8509}(50,\cdot)\)
\(\chi_{8509}(87,\cdot)\)
\(\chi_{8509}(152,\cdot)\)
\(\chi_{8509}(203,\cdot)\)
\(\chi_{8509}(221,\cdot)\)
\(\chi_{8509}(376,\cdot)\)
\(\chi_{8509}(503,\cdot)\)
\(\chi_{8509}(584,\cdot)\)
\(\chi_{8509}(660,\cdot)\)
\(\chi_{8509}(682,\cdot)\)
\(\chi_{8509}(757,\cdot)\)
\(\chi_{8509}(787,\cdot)\)
\(\chi_{8509}(950,\cdot)\)
\(\chi_{8509}(989,\cdot)\)
\(\chi_{8509}(1066,\cdot)\)
\(\chi_{8509}(1103,\cdot)\)
\(\chi_{8509}(1116,\cdot)\)
\(\chi_{8509}(1219,\cdot)\)
\(\chi_{8509}(1237,\cdot)\)
\(\chi_{8509}(1435,\cdot)\)
\(\chi_{8509}(1738,\cdot)\)
\(\chi_{8509}(1773,\cdot)\)
\(\chi_{8509}(1803,\cdot)\)
\(\chi_{8509}(1872,\cdot)\)
\(\chi_{8509}(1878,\cdot)\)
\(\chi_{8509}(2022,\cdot)\)
\(\chi_{8509}(2105,\cdot)\)
\(\chi_{8509}(2259,\cdot)\)
\(\chi_{8509}(2408,\cdot)\)
\(\chi_{8509}(2463,\cdot)\)
...
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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 };
\((2414,3686)\) → \((e\left(\frac{17}{66}\right),e\left(\frac{1}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 8509 }(221, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{317}{462}\right)\) | \(e\left(\frac{43}{462}\right)\) | \(e\left(\frac{86}{231}\right)\) | \(e\left(\frac{1}{154}\right)\) | \(e\left(\frac{60}{77}\right)\) | \(e\left(\frac{185}{462}\right)\) | \(e\left(\frac{9}{154}\right)\) | \(e\left(\frac{43}{231}\right)\) | \(e\left(\frac{160}{231}\right)\) | \(e\left(\frac{67}{154}\right)\) |
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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)
