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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(54691, base_ring=CyclotomicField(300))
M = H._module
chi = DirichletCharacter(H, M([150,275,29]))
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gp:[g,chi] = znchar(Mod(10940, 54691))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("54691.10940");
| Modulus: | \(54691\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(54691\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(300\) |
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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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\(\chi_{54691}(405,\cdot)\)
\(\chi_{54691}(678,\cdot)\)
\(\chi_{54691}(1280,\cdot)\)
\(\chi_{54691}(1826,\cdot)\)
\(\chi_{54691}(2043,\cdot)\)
\(\chi_{54691}(2169,\cdot)\)
\(\chi_{54691}(3646,\cdot)\)
\(\chi_{54691}(4115,\cdot)\)
\(\chi_{54691}(5389,\cdot)\)
\(\chi_{54691}(6194,\cdot)\)
\(\chi_{54691}(6572,\cdot)\)
\(\chi_{54691}(8301,\cdot)\)
\(\chi_{54691}(9323,\cdot)\)
\(\chi_{54691}(10940,\cdot)\)
\(\chi_{54691}(12690,\cdot)\)
\(\chi_{54691}(13271,\cdot)\)
\(\chi_{54691}(14545,\cdot)\)
\(\chi_{54691}(14783,\cdot)\)
\(\chi_{54691}(17331,\cdot)\)
\(\chi_{54691}(17821,\cdot)\)
\(\chi_{54691}(18311,\cdot)\)
\(\chi_{54691}(18570,\cdot)\)
\(\chi_{54691}(18934,\cdot)\)
\(\chi_{54691}(20131,\cdot)\)
\(\chi_{54691}(20495,\cdot)\)
\(\chi_{54691}(20754,\cdot)\)
\(\chi_{54691}(21244,\cdot)\)
\(\chi_{54691}(21734,\cdot)\)
\(\chi_{54691}(24282,\cdot)\)
\(\chi_{54691}(24520,\cdot)\)
...
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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 };
\((15627,21036,45683)\) → \((-1,e\left(\frac{11}{12}\right),e\left(\frac{29}{300}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 54691 }(10940, a) \) |
\(1\) | \(1\) | \(e\left(\frac{203}{300}\right)\) | \(e\left(\frac{83}{150}\right)\) | \(e\left(\frac{53}{150}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{23}{100}\right)\) | \(e\left(\frac{3}{100}\right)\) | \(e\left(\frac{8}{75}\right)\) | \(e\left(\frac{89}{150}\right)\) | \(e\left(\frac{89}{150}\right)\) | \(e\left(\frac{68}{75}\right)\) |
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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)
