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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(512, base_ring=CyclotomicField(128))
M = H._module
chi = DirichletCharacter(H, M([0,125]))
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gp:[g,chi] = znchar(Mod(213, 512))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("512.213");
| Modulus: | \(512\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(512\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(128\) |
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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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\(\chi_{512}(5,\cdot)\)
\(\chi_{512}(13,\cdot)\)
\(\chi_{512}(21,\cdot)\)
\(\chi_{512}(29,\cdot)\)
\(\chi_{512}(37,\cdot)\)
\(\chi_{512}(45,\cdot)\)
\(\chi_{512}(53,\cdot)\)
\(\chi_{512}(61,\cdot)\)
\(\chi_{512}(69,\cdot)\)
\(\chi_{512}(77,\cdot)\)
\(\chi_{512}(85,\cdot)\)
\(\chi_{512}(93,\cdot)\)
\(\chi_{512}(101,\cdot)\)
\(\chi_{512}(109,\cdot)\)
\(\chi_{512}(117,\cdot)\)
\(\chi_{512}(125,\cdot)\)
\(\chi_{512}(133,\cdot)\)
\(\chi_{512}(141,\cdot)\)
\(\chi_{512}(149,\cdot)\)
\(\chi_{512}(157,\cdot)\)
\(\chi_{512}(165,\cdot)\)
\(\chi_{512}(173,\cdot)\)
\(\chi_{512}(181,\cdot)\)
\(\chi_{512}(189,\cdot)\)
\(\chi_{512}(197,\cdot)\)
\(\chi_{512}(205,\cdot)\)
\(\chi_{512}(213,\cdot)\)
\(\chi_{512}(221,\cdot)\)
\(\chi_{512}(229,\cdot)\)
\(\chi_{512}(237,\cdot)\)
...
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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: |
$\Q(\zeta_{128})$ |
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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 128 polynomial (not computed) |
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sage:chi.fixed_field()
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\((511,5)\) → \((1,e\left(\frac{125}{128}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 512 }(213, a) \) |
\(1\) | \(1\) | \(e\left(\frac{23}{128}\right)\) | \(e\left(\frac{125}{128}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{1}{128}\right)\) | \(e\left(\frac{51}{128}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{59}{128}\right)\) | \(e\left(\frac{57}{128}\right)\) |
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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)
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sage:chi.gauss_sum(a)
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gp:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
