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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2584, base_ring=CyclotomicField(144))
M = H._module
chi = DirichletCharacter(H, M([0,72,99,136]))
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gp:[g,chi] = znchar(Mod(789, 2584))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2584.789");
| Modulus: | \(2584\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2584\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(144\) |
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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);
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\(\chi_{2584}(29,\cdot)\)
\(\chi_{2584}(109,\cdot)\)
\(\chi_{2584}(165,\cdot)\)
\(\chi_{2584}(173,\cdot)\)
\(\chi_{2584}(181,\cdot)\)
\(\chi_{2584}(261,\cdot)\)
\(\chi_{2584}(269,\cdot)\)
\(\chi_{2584}(317,\cdot)\)
\(\chi_{2584}(333,\cdot)\)
\(\chi_{2584}(413,\cdot)\)
\(\chi_{2584}(469,\cdot)\)
\(\chi_{2584}(573,\cdot)\)
\(\chi_{2584}(717,\cdot)\)
\(\chi_{2584}(725,\cdot)\)
\(\chi_{2584}(789,\cdot)\)
\(\chi_{2584}(813,\cdot)\)
\(\chi_{2584}(877,\cdot)\)
\(\chi_{2584}(925,\cdot)\)
\(\chi_{2584}(941,\cdot)\)
\(\chi_{2584}(1077,\cdot)\)
\(\chi_{2584}(1085,\cdot)\)
\(\chi_{2584}(1093,\cdot)\)
\(\chi_{2584}(1117,\cdot)\)
\(\chi_{2584}(1229,\cdot)\)
\(\chi_{2584}(1269,\cdot)\)
\(\chi_{2584}(1333,\cdot)\)
\(\chi_{2584}(1389,\cdot)\)
\(\chi_{2584}(1397,\cdot)\)
\(\chi_{2584}(1421,\cdot)\)
\(\chi_{2584}(1485,\cdot)\)
...
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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 };
\((647,1293,2281,1769)\) → \((1,-1,e\left(\frac{11}{16}\right),e\left(\frac{17}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 2584 }(789, a) \) |
\(1\) | \(1\) | \(e\left(\frac{67}{144}\right)\) | \(e\left(\frac{7}{144}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{67}{72}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{37}{72}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{29}{144}\right)\) | \(e\left(\frac{7}{72}\right)\) |
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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)
