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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(25721, base_ring=CyclotomicField(2992))
M = H._module
chi = DirichletCharacter(H, M([715,340]))
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gp:[g,chi] = znchar(Mod(309, 25721))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("25721.309");
| Modulus: | \(25721\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(25721\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(2992\) |
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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);
|
\(\chi_{25721}(10,\cdot)\)
\(\chi_{25721}(107,\cdot)\)
\(\chi_{25721}(109,\cdot)\)
\(\chi_{25721}(129,\cdot)\)
\(\chi_{25721}(142,\cdot)\)
\(\chi_{25721}(160,\cdot)\)
\(\chi_{25721}(173,\cdot)\)
\(\chi_{25721}(198,\cdot)\)
\(\chi_{25721}(199,\cdot)\)
\(\chi_{25721}(231,\cdot)\)
\(\chi_{25721}(250,\cdot)\)
\(\chi_{25721}(303,\cdot)\)
\(\chi_{25721}(309,\cdot)\)
\(\chi_{25721}(347,\cdot)\)
\(\chi_{25721}(377,\cdot)\)
\(\chi_{25721}(405,\cdot)\)
\(\chi_{25721}(428,\cdot)\)
\(\chi_{25721}(436,\cdot)\)
\(\chi_{25721}(454,\cdot)\)
\(\chi_{25721}(465,\cdot)\)
\(\chi_{25721}(481,\cdot)\)
\(\chi_{25721}(516,\cdot)\)
\(\chi_{25721}(524,\cdot)\)
\(\chi_{25721}(551,\cdot)\)
\(\chi_{25721}(554,\cdot)\)
\(\chi_{25721}(583,\cdot)\)
\(\chi_{25721}(602,\cdot)\)
\(\chi_{25721}(605,\cdot)\)
\(\chi_{25721}(632,\cdot)\)
\(\chi_{25721}(640,\cdot)\)
...
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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 };
\((2315,23410)\) → \((e\left(\frac{65}{272}\right),e\left(\frac{5}{44}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 25721 }(309, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{333}{1496}\right)\) | \(e\left(\frac{1055}{2992}\right)\) | \(e\left(\frac{333}{748}\right)\) | \(e\left(\frac{2031}{2992}\right)\) | \(e\left(\frac{1721}{2992}\right)\) | \(e\left(\frac{733}{2992}\right)\) | \(e\left(\frac{999}{1496}\right)\) | \(e\left(\frac{1055}{1496}\right)\) | \(e\left(\frac{2697}{2992}\right)\) | \(e\left(\frac{125}{2992}\right)\) |
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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)
