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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(24871, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([60,144,45,130]))
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gp:[g,chi] = znchar(Mod(7527, 24871))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("24871.7527");
| Modulus: | \(24871\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(24871\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(180\) |
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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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\(\chi_{24871}(268,\cdot)\)
\(\chi_{24871}(599,\cdot)\)
\(\chi_{24871}(718,\cdot)\)
\(\chi_{24871}(1857,\cdot)\)
\(\chi_{24871}(2027,\cdot)\)
\(\chi_{24871}(2979,\cdot)\)
\(\chi_{24871}(3404,\cdot)\)
\(\chi_{24871}(4118,\cdot)\)
\(\chi_{24871}(4288,\cdot)\)
\(\chi_{24871}(4722,\cdot)\)
\(\chi_{24871}(4790,\cdot)\)
\(\chi_{24871}(5240,\cdot)\)
\(\chi_{24871}(6549,\cdot)\)
\(\chi_{24871}(6983,\cdot)\)
\(\chi_{24871}(7450,\cdot)\)
\(\chi_{24871}(7527,\cdot)\)
\(\chi_{24871}(8640,\cdot)\)
\(\chi_{24871}(9244,\cdot)\)
\(\chi_{24871}(9711,\cdot)\)
\(\chi_{24871}(9762,\cdot)\)
\(\chi_{24871}(9788,\cdot)\)
\(\chi_{24871}(11071,\cdot)\)
\(\chi_{24871}(11972,\cdot)\)
\(\chi_{24871}(12049,\cdot)\)
\(\chi_{24871}(12763,\cdot)\)
\(\chi_{24871}(13766,\cdot)\)
\(\chi_{24871}(13885,\cdot)\)
\(\chi_{24871}(15024,\cdot)\)
\(\chi_{24871}(15194,\cdot)\)
\(\chi_{24871}(16146,\cdot)\)
...
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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 };
\((14213,4523,2927,11782)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{4}{5}\right),i,e\left(\frac{13}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 24871 }(7527, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{67}{180}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{121}{180}\right)\) | \(e\left(\frac{11}{180}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{67}{90}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(-i\) | \(e\left(\frac{37}{90}\right)\) |
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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)
