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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(578, base_ring=CyclotomicField(272))
M = H._module
chi = DirichletCharacter(H, M([257]))
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gp:[g,chi] = znchar(Mod(71, 578))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("578.71");
| Modulus: | \(578\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(289\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(272\) |
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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: | no, induced from \(\chi_{289}(71,\cdot)\) |
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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);
|
\(\chi_{578}(3,\cdot)\)
\(\chi_{578}(5,\cdot)\)
\(\chi_{578}(7,\cdot)\)
\(\chi_{578}(11,\cdot)\)
\(\chi_{578}(23,\cdot)\)
\(\chi_{578}(27,\cdot)\)
\(\chi_{578}(29,\cdot)\)
\(\chi_{578}(31,\cdot)\)
\(\chi_{578}(37,\cdot)\)
\(\chi_{578}(39,\cdot)\)
\(\chi_{578}(41,\cdot)\)
\(\chi_{578}(45,\cdot)\)
\(\chi_{578}(57,\cdot)\)
\(\chi_{578}(61,\cdot)\)
\(\chi_{578}(63,\cdot)\)
\(\chi_{578}(71,\cdot)\)
\(\chi_{578}(73,\cdot)\)
\(\chi_{578}(79,\cdot)\)
\(\chi_{578}(91,\cdot)\)
\(\chi_{578}(95,\cdot)\)
\(\chi_{578}(97,\cdot)\)
\(\chi_{578}(99,\cdot)\)
\(\chi_{578}(105,\cdot)\)
\(\chi_{578}(107,\cdot)\)
\(\chi_{578}(109,\cdot)\)
\(\chi_{578}(113,\cdot)\)
\(\chi_{578}(125,\cdot)\)
\(\chi_{578}(129,\cdot)\)
\(\chi_{578}(133,\cdot)\)
\(\chi_{578}(139,\cdot)\)
...
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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: |
$\Q(\zeta_{272})$ |
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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 272 polynomial (not computed) |
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sage:chi.fixed_field()
|
\(3\) → \(e\left(\frac{257}{272}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 578 }(71, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{257}{272}\right)\) | \(e\left(\frac{101}{272}\right)\) | \(e\left(\frac{123}{272}\right)\) | \(e\left(\frac{121}{136}\right)\) | \(e\left(\frac{199}{272}\right)\) | \(e\left(\frac{13}{68}\right)\) | \(e\left(\frac{43}{136}\right)\) | \(e\left(\frac{31}{136}\right)\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{191}{272}\right)\) |
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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)
