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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1875, base_ring=CyclotomicField(500))
M = H._module
chi = DirichletCharacter(H, M([250,3]))
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gp:[g,chi] = znchar(Mod(8, 1875))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1875.8");
| Modulus: | \(1875\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1875\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(500\) |
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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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\(\chi_{1875}(2,\cdot)\)
\(\chi_{1875}(8,\cdot)\)
\(\chi_{1875}(17,\cdot)\)
\(\chi_{1875}(23,\cdot)\)
\(\chi_{1875}(38,\cdot)\)
\(\chi_{1875}(47,\cdot)\)
\(\chi_{1875}(53,\cdot)\)
\(\chi_{1875}(62,\cdot)\)
\(\chi_{1875}(77,\cdot)\)
\(\chi_{1875}(83,\cdot)\)
\(\chi_{1875}(92,\cdot)\)
\(\chi_{1875}(98,\cdot)\)
\(\chi_{1875}(113,\cdot)\)
\(\chi_{1875}(122,\cdot)\)
\(\chi_{1875}(128,\cdot)\)
\(\chi_{1875}(137,\cdot)\)
\(\chi_{1875}(152,\cdot)\)
\(\chi_{1875}(158,\cdot)\)
\(\chi_{1875}(167,\cdot)\)
\(\chi_{1875}(173,\cdot)\)
\(\chi_{1875}(188,\cdot)\)
\(\chi_{1875}(197,\cdot)\)
\(\chi_{1875}(203,\cdot)\)
\(\chi_{1875}(212,\cdot)\)
\(\chi_{1875}(227,\cdot)\)
\(\chi_{1875}(233,\cdot)\)
\(\chi_{1875}(242,\cdot)\)
\(\chi_{1875}(248,\cdot)\)
\(\chi_{1875}(263,\cdot)\)
\(\chi_{1875}(272,\cdot)\)
...
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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: |
$\Q(\zeta_{500})$ |
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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 500 polynomial (not computed) |
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sage:chi.fixed_field()
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\((626,1252)\) → \((-1,e\left(\frac{3}{500}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1875 }(8, a) \) |
\(1\) | \(1\) | \(e\left(\frac{253}{500}\right)\) | \(e\left(\frac{3}{250}\right)\) | \(e\left(\frac{91}{100}\right)\) | \(e\left(\frac{259}{500}\right)\) | \(e\left(\frac{89}{250}\right)\) | \(e\left(\frac{417}{500}\right)\) | \(e\left(\frac{52}{125}\right)\) | \(e\left(\frac{3}{125}\right)\) | \(e\left(\frac{269}{500}\right)\) | \(e\left(\frac{127}{250}\right)\) |
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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)
