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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3047, base_ring=CyclotomicField(1380))
M = H._module
chi = DirichletCharacter(H, M([1104,1105]))
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gp:[g,chi] = znchar(Mod(58, 3047))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3047.58");
| Modulus: | \(3047\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3047\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1380\) |
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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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\(\chi_{3047}(5,\cdot)\)
\(\chi_{3047}(14,\cdot)\)
\(\chi_{3047}(20,\cdot)\)
\(\chi_{3047}(31,\cdot)\)
\(\chi_{3047}(53,\cdot)\)
\(\chi_{3047}(58,\cdot)\)
\(\chi_{3047}(80,\cdot)\)
\(\chi_{3047}(93,\cdot)\)
\(\chi_{3047}(97,\cdot)\)
\(\chi_{3047}(103,\cdot)\)
\(\chi_{3047}(114,\cdot)\)
\(\chi_{3047}(115,\cdot)\)
\(\chi_{3047}(119,\cdot)\)
\(\chi_{3047}(124,\cdot)\)
\(\chi_{3047}(126,\cdot)\)
\(\chi_{3047}(135,\cdot)\)
\(\chi_{3047}(137,\cdot)\)
\(\chi_{3047}(158,\cdot)\)
\(\chi_{3047}(163,\cdot)\)
\(\chi_{3047}(170,\cdot)\)
\(\chi_{3047}(174,\cdot)\)
\(\chi_{3047}(179,\cdot)\)
\(\chi_{3047}(180,\cdot)\)
\(\chi_{3047}(181,\cdot)\)
\(\chi_{3047}(212,\cdot)\)
\(\chi_{3047}(224,\cdot)\)
\(\chi_{3047}(234,\cdot)\)
\(\chi_{3047}(246,\cdot)\)
\(\chi_{3047}(257,\cdot)\)
\(\chi_{3047}(291,\cdot)\)
...
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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 };
\((1663,2498)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{221}{276}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 3047 }(58, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{233}{460}\right)\) | \(e\left(\frac{323}{345}\right)\) | \(e\left(\frac{3}{230}\right)\) | \(e\left(\frac{1}{1380}\right)\) | \(e\left(\frac{611}{1380}\right)\) | \(e\left(\frac{149}{690}\right)\) | \(e\left(\frac{239}{460}\right)\) | \(e\left(\frac{301}{345}\right)\) | \(e\left(\frac{35}{69}\right)\) | \(e\left(\frac{131}{138}\right)\) |
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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)
