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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(343, base_ring=CyclotomicField(294))
M = H._module
chi = DirichletCharacter(H, M([220]))
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gp:[g,chi] = znchar(Mod(102, 343))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("343.102");
| Modulus: | \(343\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(343\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(147\) |
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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);
|
\(\chi_{343}(2,\cdot)\)
\(\chi_{343}(4,\cdot)\)
\(\chi_{343}(9,\cdot)\)
\(\chi_{343}(11,\cdot)\)
\(\chi_{343}(16,\cdot)\)
\(\chi_{343}(23,\cdot)\)
\(\chi_{343}(25,\cdot)\)
\(\chi_{343}(32,\cdot)\)
\(\chi_{343}(37,\cdot)\)
\(\chi_{343}(39,\cdot)\)
\(\chi_{343}(44,\cdot)\)
\(\chi_{343}(46,\cdot)\)
\(\chi_{343}(51,\cdot)\)
\(\chi_{343}(53,\cdot)\)
\(\chi_{343}(58,\cdot)\)
\(\chi_{343}(60,\cdot)\)
\(\chi_{343}(65,\cdot)\)
\(\chi_{343}(72,\cdot)\)
\(\chi_{343}(74,\cdot)\)
\(\chi_{343}(81,\cdot)\)
\(\chi_{343}(86,\cdot)\)
\(\chi_{343}(88,\cdot)\)
\(\chi_{343}(93,\cdot)\)
\(\chi_{343}(95,\cdot)\)
\(\chi_{343}(100,\cdot)\)
\(\chi_{343}(102,\cdot)\)
\(\chi_{343}(107,\cdot)\)
\(\chi_{343}(109,\cdot)\)
\(\chi_{343}(114,\cdot)\)
\(\chi_{343}(121,\cdot)\)
...
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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: |
$\Q(\zeta_{147})$ |
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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 147 polynomial (not computed) |
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sage:chi.fixed_field()
|
\(3\) → \(e\left(\frac{110}{147}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 343 }(102, a) \) |
\(1\) | \(1\) | \(e\left(\frac{25}{147}\right)\) | \(e\left(\frac{110}{147}\right)\) | \(e\left(\frac{50}{147}\right)\) | \(e\left(\frac{103}{147}\right)\) | \(e\left(\frac{45}{49}\right)\) | \(e\left(\frac{25}{49}\right)\) | \(e\left(\frac{73}{147}\right)\) | \(e\left(\frac{128}{147}\right)\) | \(e\left(\frac{95}{147}\right)\) | \(e\left(\frac{13}{147}\right)\) |
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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)
