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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(18943, base_ring=CyclotomicField(498))
M = H._module
chi = DirichletCharacter(H, M([83,79]))
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gp:[g,chi] = znchar(Mod(1718, 18943))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("18943.1718");
| Modulus: | \(18943\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(18943\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(498\) |
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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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\(\chi_{18943}(126,\cdot)\)
\(\chi_{18943}(255,\cdot)\)
\(\chi_{18943}(449,\cdot)\)
\(\chi_{18943}(468,\cdot)\)
\(\chi_{18943}(563,\cdot)\)
\(\chi_{18943}(635,\cdot)\)
\(\chi_{18943}(730,\cdot)\)
\(\chi_{18943}(734,\cdot)\)
\(\chi_{18943}(848,\cdot)\)
\(\chi_{18943}(939,\cdot)\)
\(\chi_{18943}(1019,\cdot)\)
\(\chi_{18943}(1091,\cdot)\)
\(\chi_{18943}(1224,\cdot)\)
\(\chi_{18943}(1452,\cdot)\)
\(\chi_{18943}(1475,\cdot)\)
\(\chi_{18943}(1566,\cdot)\)
\(\chi_{18943}(1585,\cdot)\)
\(\chi_{18943}(1718,\cdot)\)
\(\chi_{18943}(2231,\cdot)\)
\(\chi_{18943}(2421,\cdot)\)
\(\chi_{18943}(2763,\cdot)\)
\(\chi_{18943}(2820,\cdot)\)
\(\chi_{18943}(2972,\cdot)\)
\(\chi_{18943}(3048,\cdot)\)
\(\chi_{18943}(3067,\cdot)\)
\(\chi_{18943}(3128,\cdot)\)
\(\chi_{18943}(3181,\cdot)\)
\(\chi_{18943}(3280,\cdot)\)
\(\chi_{18943}(3394,\cdot)\)
\(\chi_{18943}(3466,\cdot)\)
...
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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 };
\((16950,11971)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{79}{498}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 18943 }(1718, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{13}{249}\right)\) | \(e\left(\frac{59}{498}\right)\) | \(e\left(\frac{26}{249}\right)\) | \(e\left(\frac{215}{498}\right)\) | \(e\left(\frac{85}{498}\right)\) | \(e\left(\frac{79}{498}\right)\) | \(e\left(\frac{13}{83}\right)\) | \(e\left(\frac{59}{249}\right)\) | \(e\left(\frac{241}{498}\right)\) | \(e\left(\frac{301}{498}\right)\) |
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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)
