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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(767, base_ring=CyclotomicField(174))
M = H._module
chi = DirichletCharacter(H, M([58,33]))
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gp:[g,chi] = znchar(Mod(42, 767))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("767.42");
| Modulus: | \(767\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(767\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(174\) |
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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: | odd |
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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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\(\chi_{767}(42,\cdot)\)
\(\chi_{767}(55,\cdot)\)
\(\chi_{767}(61,\cdot)\)
\(\chi_{767}(113,\cdot)\)
\(\chi_{767}(120,\cdot)\)
\(\chi_{767}(126,\cdot)\)
\(\chi_{767}(152,\cdot)\)
\(\chi_{767}(165,\cdot)\)
\(\chi_{767}(172,\cdot)\)
\(\chi_{767}(185,\cdot)\)
\(\chi_{767}(191,\cdot)\)
\(\chi_{767}(211,\cdot)\)
\(\chi_{767}(217,\cdot)\)
\(\chi_{767}(224,\cdot)\)
\(\chi_{767}(250,\cdot)\)
\(\chi_{767}(269,\cdot)\)
\(\chi_{767}(276,\cdot)\)
\(\chi_{767}(308,\cdot)\)
\(\chi_{767}(328,\cdot)\)
\(\chi_{767}(334,\cdot)\)
\(\chi_{767}(347,\cdot)\)
\(\chi_{767}(360,\cdot)\)
\(\chi_{767}(367,\cdot)\)
\(\chi_{767}(386,\cdot)\)
\(\chi_{767}(393,\cdot)\)
\(\chi_{767}(406,\cdot)\)
\(\chi_{767}(419,\cdot)\)
\(\chi_{767}(445,\cdot)\)
\(\chi_{767}(451,\cdot)\)
\(\chi_{767}(490,\cdot)\)
...
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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: |
$\Q(\zeta_{87})$ |
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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 174 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((119,651)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{11}{58}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 767 }(42, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{91}{174}\right)\) | \(e\left(\frac{71}{87}\right)\) | \(e\left(\frac{4}{87}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{59}{174}\right)\) | \(e\left(\frac{7}{87}\right)\) | \(e\left(\frac{33}{58}\right)\) | \(e\left(\frac{55}{87}\right)\) | \(e\left(\frac{115}{174}\right)\) | \(e\left(\frac{13}{174}\right)\) |
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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)
