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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(981, base_ring=CyclotomicField(108))
M = H._module
chi = DirichletCharacter(H, M([36,55]))
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gp:[g,chi] = znchar(Mod(103, 981))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("981.103");
| Modulus: | \(981\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(981\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(108\) |
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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);
|
\(\chi_{981}(13,\cdot)\)
\(\chi_{981}(40,\cdot)\)
\(\chi_{981}(58,\cdot)\)
\(\chi_{981}(70,\cdot)\)
\(\chi_{981}(79,\cdot)\)
\(\chi_{981}(85,\cdot)\)
\(\chi_{981}(103,\cdot)\)
\(\chi_{981}(151,\cdot)\)
\(\chi_{981}(166,\cdot)\)
\(\chi_{981}(268,\cdot)\)
\(\chi_{981}(274,\cdot)\)
\(\chi_{981}(277,\cdot)\)
\(\chi_{981}(337,\cdot)\)
\(\chi_{981}(394,\cdot)\)
\(\chi_{981}(466,\cdot)\)
\(\chi_{981}(475,\cdot)\)
\(\chi_{981}(535,\cdot)\)
\(\chi_{981}(556,\cdot)\)
\(\chi_{981}(592,\cdot)\)
\(\chi_{981}(598,\cdot)\)
\(\chi_{981}(610,\cdot)\)
\(\chi_{981}(691,\cdot)\)
\(\chi_{981}(706,\cdot)\)
\(\chi_{981}(745,\cdot)\)
\(\chi_{981}(769,\cdot)\)
\(\chi_{981}(781,\cdot)\)
\(\chi_{981}(787,\cdot)\)
\(\chi_{981}(814,\cdot)\)
\(\chi_{981}(832,\cdot)\)
\(\chi_{981}(835,\cdot)\)
...
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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 };
\((110,442)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{55}{108}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 981 }(103, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{79}{108}\right)\) | \(e\left(\frac{65}{108}\right)\) | \(e\left(\frac{85}{108}\right)\) | \(e\left(\frac{7}{108}\right)\) | \(e\left(\frac{4}{9}\right)\) |
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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)
