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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(967, base_ring=CyclotomicField(966))
M = H._module
chi = DirichletCharacter(H, M([374]))
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gp:[g,chi] = znchar(Mod(2, 967))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("967.2");
| Modulus: | \(967\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(967\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(483\) |
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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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\(\chi_{967}(2,\cdot)\)
\(\chi_{967}(4,\cdot)\)
\(\chi_{967}(16,\cdot)\)
\(\chi_{967}(18,\cdot)\)
\(\chi_{967}(21,\cdot)\)
\(\chi_{967}(22,\cdot)\)
\(\chi_{967}(25,\cdot)\)
\(\chi_{967}(31,\cdot)\)
\(\chi_{967}(32,\cdot)\)
\(\chi_{967}(34,\cdot)\)
\(\chi_{967}(35,\cdot)\)
\(\chi_{967}(36,\cdot)\)
\(\chi_{967}(44,\cdot)\)
\(\chi_{967}(49,\cdot)\)
\(\chi_{967}(50,\cdot)\)
\(\chi_{967}(57,\cdot)\)
\(\chi_{967}(59,\cdot)\)
\(\chi_{967}(60,\cdot)\)
\(\chi_{967}(65,\cdot)\)
\(\chi_{967}(70,\cdot)\)
\(\chi_{967}(83,\cdot)\)
\(\chi_{967}(84,\cdot)\)
\(\chi_{967}(91,\cdot)\)
\(\chi_{967}(98,\cdot)\)
\(\chi_{967}(101,\cdot)\)
\(\chi_{967}(103,\cdot)\)
\(\chi_{967}(106,\cdot)\)
\(\chi_{967}(111,\cdot)\)
\(\chi_{967}(114,\cdot)\)
\(\chi_{967}(115,\cdot)\)
...
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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: |
$\Q(\zeta_{483})$ |
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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 483 polynomial (not computed) |
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sage:chi.fixed_field()
|
\(5\) → \(e\left(\frac{187}{483}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 967 }(2, a) \) |
\(1\) | \(1\) | \(e\left(\frac{386}{483}\right)\) | \(e\left(\frac{122}{161}\right)\) | \(e\left(\frac{289}{483}\right)\) | \(e\left(\frac{187}{483}\right)\) | \(e\left(\frac{269}{483}\right)\) | \(e\left(\frac{454}{483}\right)\) | \(e\left(\frac{64}{161}\right)\) | \(e\left(\frac{83}{161}\right)\) | \(e\left(\frac{30}{161}\right)\) | \(e\left(\frac{129}{161}\right)\) |
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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)
