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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(11725, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([99,220,120]))
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gp:[g,chi] = znchar(Mod(3364, 11725))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("11725.3364");
| Modulus: | \(11725\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(11725\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(330\) |
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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);
|
\(\chi_{11725}(9,\cdot)\)
\(\chi_{11725}(359,\cdot)\)
\(\chi_{11725}(394,\cdot)\)
\(\chi_{11725}(464,\cdot)\)
\(\chi_{11725}(494,\cdot)\)
\(\chi_{11725}(844,\cdot)\)
\(\chi_{11725}(1019,\cdot)\)
\(\chi_{11725}(1094,\cdot)\)
\(\chi_{11725}(1164,\cdot)\)
\(\chi_{11725}(1404,\cdot)\)
\(\chi_{11725}(1514,\cdot)\)
\(\chi_{11725}(1684,\cdot)\)
\(\chi_{11725}(1689,\cdot)\)
\(\chi_{11725}(2034,\cdot)\)
\(\chi_{11725}(2069,\cdot)\)
\(\chi_{11725}(2139,\cdot)\)
\(\chi_{11725}(2354,\cdot)\)
\(\chi_{11725}(2494,\cdot)\)
\(\chi_{11725}(2704,\cdot)\)
\(\chi_{11725}(2739,\cdot)\)
\(\chi_{11725}(2769,\cdot)\)
\(\chi_{11725}(2809,\cdot)\)
\(\chi_{11725}(2839,\cdot)\)
\(\chi_{11725}(3189,\cdot)\)
\(\chi_{11725}(3364,\cdot)\)
\(\chi_{11725}(3439,\cdot)\)
\(\chi_{11725}(3509,\cdot)\)
\(\chi_{11725}(3859,\cdot)\)
\(\chi_{11725}(4029,\cdot)\)
\(\chi_{11725}(4034,\cdot)\)
...
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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 };
\((1877,1676,3151)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{2}{3}\right),e\left(\frac{4}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 11725 }(3364, a) \) |
\(1\) | \(1\) | \(e\left(\frac{329}{330}\right)\) | \(e\left(\frac{313}{330}\right)\) | \(e\left(\frac{164}{165}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{148}{165}\right)\) | \(e\left(\frac{152}{165}\right)\) | \(e\left(\frac{311}{330}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{163}{165}\right)\) |
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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)
