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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(437, base_ring=CyclotomicField(198))
M = H._module
chi = DirichletCharacter(H, M([44,36]))
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gp:[g,chi] = znchar(Mod(73, 437))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("437.73");
| Modulus: | \(437\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(437\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(99\) |
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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);
|
\(\chi_{437}(4,\cdot)\)
\(\chi_{437}(6,\cdot)\)
\(\chi_{437}(9,\cdot)\)
\(\chi_{437}(16,\cdot)\)
\(\chi_{437}(25,\cdot)\)
\(\chi_{437}(35,\cdot)\)
\(\chi_{437}(36,\cdot)\)
\(\chi_{437}(54,\cdot)\)
\(\chi_{437}(55,\cdot)\)
\(\chi_{437}(62,\cdot)\)
\(\chi_{437}(73,\cdot)\)
\(\chi_{437}(81,\cdot)\)
\(\chi_{437}(82,\cdot)\)
\(\chi_{437}(85,\cdot)\)
\(\chi_{437}(100,\cdot)\)
\(\chi_{437}(101,\cdot)\)
\(\chi_{437}(104,\cdot)\)
\(\chi_{437}(118,\cdot)\)
\(\chi_{437}(119,\cdot)\)
\(\chi_{437}(123,\cdot)\)
\(\chi_{437}(131,\cdot)\)
\(\chi_{437}(142,\cdot)\)
\(\chi_{437}(150,\cdot)\)
\(\chi_{437}(156,\cdot)\)
\(\chi_{437}(169,\cdot)\)
\(\chi_{437}(177,\cdot)\)
\(\chi_{437}(187,\cdot)\)
\(\chi_{437}(188,\cdot)\)
\(\chi_{437}(196,\cdot)\)
\(\chi_{437}(213,\cdot)\)
...
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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 };
\((116,419)\) → \((e\left(\frac{2}{9}\right),e\left(\frac{2}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 437 }(73, a) \) |
\(1\) | \(1\) | \(e\left(\frac{58}{99}\right)\) | \(e\left(\frac{79}{99}\right)\) | \(e\left(\frac{17}{99}\right)\) | \(e\left(\frac{73}{99}\right)\) | \(e\left(\frac{38}{99}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{59}{99}\right)\) | \(e\left(\frac{32}{99}\right)\) | \(e\left(\frac{10}{33}\right)\) |
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sage:chi(x)
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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)
