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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(32851, base_ring=CyclotomicField(684))
M = H._module
chi = DirichletCharacter(H, M([570,513,394]))
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gp:[g,chi] = znchar(Mod(4646, 32851))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("32851.4646");
| Modulus: | \(32851\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(32851\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(684\) |
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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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\(\chi_{32851}(642,\cdot)\)
\(\chi_{32851}(775,\cdot)\)
\(\chi_{32851}(801,\cdot)\)
\(\chi_{32851}(850,\cdot)\)
\(\chi_{32851}(983,\cdot)\)
\(\chi_{32851}(1123,\cdot)\)
\(\chi_{32851}(1188,\cdot)\)
\(\chi_{32851}(1256,\cdot)\)
\(\chi_{32851}(1321,\cdot)\)
\(\chi_{32851}(1552,\cdot)\)
\(\chi_{32851}(1685,\cdot)\)
\(\chi_{32851}(2371,\cdot)\)
\(\chi_{32851}(2397,\cdot)\)
\(\chi_{32851}(2504,\cdot)\)
\(\chi_{32851}(2530,\cdot)\)
\(\chi_{32851}(2579,\cdot)\)
\(\chi_{32851}(2712,\cdot)\)
\(\chi_{32851}(2852,\cdot)\)
\(\chi_{32851}(2917,\cdot)\)
\(\chi_{32851}(2985,\cdot)\)
\(\chi_{32851}(3050,\cdot)\)
\(\chi_{32851}(3281,\cdot)\)
\(\chi_{32851}(3414,\cdot)\)
\(\chi_{32851}(4100,\cdot)\)
\(\chi_{32851}(4126,\cdot)\)
\(\chi_{32851}(4259,\cdot)\)
\(\chi_{32851}(4308,\cdot)\)
\(\chi_{32851}(4441,\cdot)\)
\(\chi_{32851}(4581,\cdot)\)
\(\chi_{32851}(4646,\cdot)\)
...
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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 };
\((14080,20217,22023)\) → \((e\left(\frac{5}{6}\right),-i,e\left(\frac{197}{342}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 32851 }(4646, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{679}{684}\right)\) | \(e\left(\frac{154}{171}\right)\) | \(e\left(\frac{337}{342}\right)\) | \(e\left(\frac{379}{684}\right)\) | \(e\left(\frac{611}{684}\right)\) | \(e\left(\frac{223}{228}\right)\) | \(e\left(\frac{137}{171}\right)\) | \(e\left(\frac{187}{342}\right)\) | \(e\left(\frac{77}{228}\right)\) | \(e\left(\frac{101}{114}\right)\) |
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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)
