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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1143, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([105,23]))
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gp:[g,chi] = znchar(Mod(464, 1143))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1143.464");
| Modulus: | \(1143\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1143\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(126\) |
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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: | even |
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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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\(\chi_{1143}(56,\cdot)\)
\(\chi_{1143}(65,\cdot)\)
\(\chi_{1143}(86,\cdot)\)
\(\chi_{1143}(92,\cdot)\)
\(\chi_{1143}(101,\cdot)\)
\(\chi_{1143}(173,\cdot)\)
\(\chi_{1143}(185,\cdot)\)
\(\chi_{1143}(212,\cdot)\)
\(\chi_{1143}(293,\cdot)\)
\(\chi_{1143}(347,\cdot)\)
\(\chi_{1143}(410,\cdot)\)
\(\chi_{1143}(464,\cdot)\)
\(\chi_{1143}(491,\cdot)\)
\(\chi_{1143}(515,\cdot)\)
\(\chi_{1143}(563,\cdot)\)
\(\chi_{1143}(599,\cdot)\)
\(\chi_{1143}(605,\cdot)\)
\(\chi_{1143}(617,\cdot)\)
\(\chi_{1143}(626,\cdot)\)
\(\chi_{1143}(641,\cdot)\)
\(\chi_{1143}(680,\cdot)\)
\(\chi_{1143}(731,\cdot)\)
\(\chi_{1143}(749,\cdot)\)
\(\chi_{1143}(776,\cdot)\)
\(\chi_{1143}(785,\cdot)\)
\(\chi_{1143}(815,\cdot)\)
\(\chi_{1143}(878,\cdot)\)
\(\chi_{1143}(932,\cdot)\)
\(\chi_{1143}(956,\cdot)\)
\(\chi_{1143}(995,\cdot)\)
...
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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 };
\((128,892)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{23}{126}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 1143 }(464, a) \) |
\(1\) | \(1\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{41}{126}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{31}{126}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{19}{21}\right)\) |
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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)
