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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(15517, base_ring=CyclotomicField(7598))
M = H._module
chi = DirichletCharacter(H, M([393,1334]))
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gp:[g,chi] = znchar(Mod(8, 15517))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("15517.8");
| Modulus: | \(15517\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(15517\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(7598\) |
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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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\(\chi_{15517}(2,\cdot)\)
\(\chi_{15517}(6,\cdot)\)
\(\chi_{15517}(8,\cdot)\)
\(\chi_{15517}(11,\cdot)\)
\(\chi_{15517}(13,\cdot)\)
\(\chi_{15517}(18,\cdot)\)
\(\chi_{15517}(23,\cdot)\)
\(\chi_{15517}(24,\cdot)\)
\(\chi_{15517}(31,\cdot)\)
\(\chi_{15517}(32,\cdot)\)
\(\chi_{15517}(33,\cdot)\)
\(\chi_{15517}(34,\cdot)\)
\(\chi_{15517}(37,\cdot)\)
\(\chi_{15517}(39,\cdot)\)
\(\chi_{15517}(43,\cdot)\)
\(\chi_{15517}(44,\cdot)\)
\(\chi_{15517}(50,\cdot)\)
\(\chi_{15517}(52,\cdot)\)
\(\chi_{15517}(54,\cdot)\)
\(\chi_{15517}(61,\cdot)\)
\(\chi_{15517}(69,\cdot)\)
\(\chi_{15517}(70,\cdot)\)
\(\chi_{15517}(72,\cdot)\)
\(\chi_{15517}(83,\cdot)\)
\(\chi_{15517}(89,\cdot)\)
\(\chi_{15517}(92,\cdot)\)
\(\chi_{15517}(93,\cdot)\)
\(\chi_{15517}(96,\cdot)\)
\(\chi_{15517}(98,\cdot)\)
\(\chi_{15517}(99,\cdot)\)
...
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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 };
| Field of values: |
$\Q(\zeta_{3799})$ |
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sage:CyclotomicField(chi.multiplicative_order())
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gp:nfinit(polcyclo(charorder(g,chi)))
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magma:CyclotomicField(Order(chi));
|
| Fixed field: |
Number field defined by a degree 7598 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((9206,9736)\) → \((e\left(\frac{3}{58}\right),e\left(\frac{23}{131}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 15517 }(8, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{3119}{7598}\right)\) | \(e\left(\frac{1386}{3799}\right)\) | \(e\left(\frac{3119}{3799}\right)\) | \(e\left(\frac{1846}{3799}\right)\) | \(e\left(\frac{5891}{7598}\right)\) | \(e\left(\frac{3044}{3799}\right)\) | \(e\left(\frac{1759}{7598}\right)\) | \(e\left(\frac{2772}{3799}\right)\) | \(e\left(\frac{6811}{7598}\right)\) | \(e\left(\frac{3329}{7598}\right)\) |
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sage:chi(x) # x integer
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gp:chareval(g,chi,x) \\ x integer, value in Q/Z
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magma:chi(x)
