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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6381, base_ring=CyclotomicField(236))
M = H._module
chi = DirichletCharacter(H, M([118,149]))
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gp:[g,chi] = znchar(Mod(575, 6381))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6381.575");
| Modulus: | \(6381\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2127\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(236\) |
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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: | no, induced from \(\chi_{2127}(575,\cdot)\) |
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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);
|
\(\chi_{6381}(8,\cdot)\)
\(\chi_{6381}(53,\cdot)\)
\(\chi_{6381}(98,\cdot)\)
\(\chi_{6381}(107,\cdot)\)
\(\chi_{6381}(134,\cdot)\)
\(\chi_{6381}(197,\cdot)\)
\(\chi_{6381}(206,\cdot)\)
\(\chi_{6381}(224,\cdot)\)
\(\chi_{6381}(242,\cdot)\)
\(\chi_{6381}(260,\cdot)\)
\(\chi_{6381}(287,\cdot)\)
\(\chi_{6381}(314,\cdot)\)
\(\chi_{6381}(395,\cdot)\)
\(\chi_{6381}(422,\cdot)\)
\(\chi_{6381}(449,\cdot)\)
\(\chi_{6381}(467,\cdot)\)
\(\chi_{6381}(485,\cdot)\)
\(\chi_{6381}(503,\cdot)\)
\(\chi_{6381}(512,\cdot)\)
\(\chi_{6381}(575,\cdot)\)
\(\chi_{6381}(602,\cdot)\)
\(\chi_{6381}(611,\cdot)\)
\(\chi_{6381}(656,\cdot)\)
\(\chi_{6381}(701,\cdot)\)
\(\chi_{6381}(782,\cdot)\)
\(\chi_{6381}(818,\cdot)\)
\(\chi_{6381}(863,\cdot)\)
\(\chi_{6381}(899,\cdot)\)
\(\chi_{6381}(935,\cdot)\)
\(\chi_{6381}(953,\cdot)\)
...
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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 };
\((2837,3547)\) → \((-1,e\left(\frac{149}{236}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 6381 }(575, a) \) |
\(1\) | \(1\) | \(e\left(\frac{31}{236}\right)\) | \(e\left(\frac{31}{118}\right)\) | \(e\left(\frac{47}{59}\right)\) | \(e\left(\frac{49}{59}\right)\) | \(e\left(\frac{93}{236}\right)\) | \(e\left(\frac{219}{236}\right)\) | \(e\left(\frac{30}{59}\right)\) | \(e\left(\frac{217}{236}\right)\) | \(e\left(\frac{227}{236}\right)\) | \(e\left(\frac{31}{59}\right)\) |
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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)
