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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2285, base_ring=CyclotomicField(456))
M = H._module
chi = DirichletCharacter(H, M([114,29]))
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gp:[g,chi] = znchar(Mod(317, 2285))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2285.317");
| Modulus: | \(2285\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2285\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(456\) |
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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);
|
\(\chi_{2285}(33,\cdot)\)
\(\chi_{2285}(52,\cdot)\)
\(\chi_{2285}(62,\cdot)\)
\(\chi_{2285}(77,\cdot)\)
\(\chi_{2285}(82,\cdot)\)
\(\chi_{2285}(92,\cdot)\)
\(\chi_{2285}(118,\cdot)\)
\(\chi_{2285}(142,\cdot)\)
\(\chi_{2285}(173,\cdot)\)
\(\chi_{2285}(178,\cdot)\)
\(\chi_{2285}(198,\cdot)\)
\(\chi_{2285}(212,\cdot)\)
\(\chi_{2285}(258,\cdot)\)
\(\chi_{2285}(273,\cdot)\)
\(\chi_{2285}(277,\cdot)\)
\(\chi_{2285}(278,\cdot)\)
\(\chi_{2285}(293,\cdot)\)
\(\chi_{2285}(298,\cdot)\)
\(\chi_{2285}(303,\cdot)\)
\(\chi_{2285}(312,\cdot)\)
\(\chi_{2285}(317,\cdot)\)
\(\chi_{2285}(333,\cdot)\)
\(\chi_{2285}(337,\cdot)\)
\(\chi_{2285}(353,\cdot)\)
\(\chi_{2285}(362,\cdot)\)
\(\chi_{2285}(388,\cdot)\)
\(\chi_{2285}(412,\cdot)\)
\(\chi_{2285}(418,\cdot)\)
\(\chi_{2285}(422,\cdot)\)
\(\chi_{2285}(427,\cdot)\)
...
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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 };
\((1372,1841)\) → \((i,e\left(\frac{29}{456}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 2285 }(317, a) \) |
\(1\) | \(1\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{41}{114}\right)\) | \(e\left(\frac{11}{228}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{119}{152}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{371}{456}\right)\) |
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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)
