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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([3406,2726]))
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gp:[g,chi] = znchar(Mod(22, 15517))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("15517.22");
| 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: | \(3799\) |
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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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\(\chi_{15517}(3,\cdot)\)
\(\chi_{15517}(4,\cdot)\)
\(\chi_{15517}(9,\cdot)\)
\(\chi_{15517}(12,\cdot)\)
\(\chi_{15517}(16,\cdot)\)
\(\chi_{15517}(17,\cdot)\)
\(\chi_{15517}(22,\cdot)\)
\(\chi_{15517}(25,\cdot)\)
\(\chi_{15517}(26,\cdot)\)
\(\chi_{15517}(27,\cdot)\)
\(\chi_{15517}(35,\cdot)\)
\(\chi_{15517}(36,\cdot)\)
\(\chi_{15517}(46,\cdot)\)
\(\chi_{15517}(48,\cdot)\)
\(\chi_{15517}(49,\cdot)\)
\(\chi_{15517}(51,\cdot)\)
\(\chi_{15517}(62,\cdot)\)
\(\chi_{15517}(64,\cdot)\)
\(\chi_{15517}(66,\cdot)\)
\(\chi_{15517}(68,\cdot)\)
\(\chi_{15517}(74,\cdot)\)
\(\chi_{15517}(75,\cdot)\)
\(\chi_{15517}(78,\cdot)\)
\(\chi_{15517}(81,\cdot)\)
\(\chi_{15517}(86,\cdot)\)
\(\chi_{15517}(88,\cdot)\)
\(\chi_{15517}(95,\cdot)\)
\(\chi_{15517}(100,\cdot)\)
\(\chi_{15517}(104,\cdot)\)
\(\chi_{15517}(105,\cdot)\)
...
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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 3799 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((9206,9736)\) → \((e\left(\frac{13}{29}\right),e\left(\frac{47}{131}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 15517 }(22, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2341}{3799}\right)\) | \(e\left(\frac{1340}{3799}\right)\) | \(e\left(\frac{883}{3799}\right)\) | \(e\left(\frac{184}{3799}\right)\) | \(e\left(\frac{3681}{3799}\right)\) | \(e\left(\frac{1567}{3799}\right)\) | \(e\left(\frac{3224}{3799}\right)\) | \(e\left(\frac{2680}{3799}\right)\) | \(e\left(\frac{2525}{3799}\right)\) | \(e\left(\frac{2903}{3799}\right)\) |
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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)
