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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4232, base_ring=CyclotomicField(506))
M = H._module
chi = DirichletCharacter(H, M([0,253,150]))
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gp:[g,chi] = znchar(Mod(29, 4232))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4232.29");
| Modulus: | \(4232\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4232\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(506\) |
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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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\(\chi_{4232}(13,\cdot)\)
\(\chi_{4232}(29,\cdot)\)
\(\chi_{4232}(77,\cdot)\)
\(\chi_{4232}(85,\cdot)\)
\(\chi_{4232}(101,\cdot)\)
\(\chi_{4232}(117,\cdot)\)
\(\chi_{4232}(133,\cdot)\)
\(\chi_{4232}(141,\cdot)\)
\(\chi_{4232}(165,\cdot)\)
\(\chi_{4232}(173,\cdot)\)
\(\chi_{4232}(197,\cdot)\)
\(\chi_{4232}(213,\cdot)\)
\(\chi_{4232}(261,\cdot)\)
\(\chi_{4232}(269,\cdot)\)
\(\chi_{4232}(285,\cdot)\)
\(\chi_{4232}(301,\cdot)\)
\(\chi_{4232}(317,\cdot)\)
\(\chi_{4232}(325,\cdot)\)
\(\chi_{4232}(349,\cdot)\)
\(\chi_{4232}(357,\cdot)\)
\(\chi_{4232}(381,\cdot)\)
\(\chi_{4232}(397,\cdot)\)
\(\chi_{4232}(445,\cdot)\)
\(\chi_{4232}(453,\cdot)\)
\(\chi_{4232}(469,\cdot)\)
\(\chi_{4232}(485,\cdot)\)
\(\chi_{4232}(509,\cdot)\)
\(\chi_{4232}(533,\cdot)\)
\(\chi_{4232}(541,\cdot)\)
\(\chi_{4232}(565,\cdot)\)
...
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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: |
$\Q(\zeta_{253})$ |
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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 506 polynomial (not computed) |
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sage:chi.fixed_field()
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\((3175,2117,2121)\) → \((1,-1,e\left(\frac{75}{253}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 4232 }(29, a) \) |
\(1\) | \(1\) | \(e\left(\frac{123}{506}\right)\) | \(e\left(\frac{403}{506}\right)\) | \(e\left(\frac{61}{253}\right)\) | \(e\left(\frac{123}{253}\right)\) | \(e\left(\frac{305}{506}\right)\) | \(e\left(\frac{417}{506}\right)\) | \(e\left(\frac{10}{253}\right)\) | \(e\left(\frac{151}{253}\right)\) | \(e\left(\frac{215}{506}\right)\) | \(e\left(\frac{245}{506}\right)\) |
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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)
