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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(847, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([55,96]))
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gp:[g,chi] = znchar(Mod(59, 847))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("847.59");
| Modulus: | \(847\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(847\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(330\) |
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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);
|
\(\chi_{847}(5,\cdot)\)
\(\chi_{847}(26,\cdot)\)
\(\chi_{847}(31,\cdot)\)
\(\chi_{847}(38,\cdot)\)
\(\chi_{847}(47,\cdot)\)
\(\chi_{847}(59,\cdot)\)
\(\chi_{847}(75,\cdot)\)
\(\chi_{847}(80,\cdot)\)
\(\chi_{847}(82,\cdot)\)
\(\chi_{847}(103,\cdot)\)
\(\chi_{847}(108,\cdot)\)
\(\chi_{847}(115,\cdot)\)
\(\chi_{847}(136,\cdot)\)
\(\chi_{847}(152,\cdot)\)
\(\chi_{847}(157,\cdot)\)
\(\chi_{847}(159,\cdot)\)
\(\chi_{847}(180,\cdot)\)
\(\chi_{847}(185,\cdot)\)
\(\chi_{847}(192,\cdot)\)
\(\chi_{847}(201,\cdot)\)
\(\chi_{847}(213,\cdot)\)
\(\chi_{847}(229,\cdot)\)
\(\chi_{847}(234,\cdot)\)
\(\chi_{847}(236,\cdot)\)
\(\chi_{847}(257,\cdot)\)
\(\chi_{847}(262,\cdot)\)
\(\chi_{847}(278,\cdot)\)
\(\chi_{847}(290,\cdot)\)
\(\chi_{847}(306,\cdot)\)
\(\chi_{847}(311,\cdot)\)
...
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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 };
\((122,365)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{16}{55}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 847 }(59, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{103}{165}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{41}{165}\right)\) | \(e\left(\frac{119}{330}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{97}{110}\right)\) |
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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)
