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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(507, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([0,1]))
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gp:[g,chi] = znchar(Mod(340, 507))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("507.340");
| Modulus: | \(507\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(169\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(156\) |
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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: | no, induced from \(\chi_{169}(2,\cdot)\) |
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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);
|
\(\chi_{507}(7,\cdot)\)
\(\chi_{507}(28,\cdot)\)
\(\chi_{507}(37,\cdot)\)
\(\chi_{507}(46,\cdot)\)
\(\chi_{507}(58,\cdot)\)
\(\chi_{507}(67,\cdot)\)
\(\chi_{507}(76,\cdot)\)
\(\chi_{507}(85,\cdot)\)
\(\chi_{507}(97,\cdot)\)
\(\chi_{507}(106,\cdot)\)
\(\chi_{507}(115,\cdot)\)
\(\chi_{507}(124,\cdot)\)
\(\chi_{507}(136,\cdot)\)
\(\chi_{507}(145,\cdot)\)
\(\chi_{507}(154,\cdot)\)
\(\chi_{507}(163,\cdot)\)
\(\chi_{507}(175,\cdot)\)
\(\chi_{507}(184,\cdot)\)
\(\chi_{507}(193,\cdot)\)
\(\chi_{507}(202,\cdot)\)
\(\chi_{507}(214,\cdot)\)
\(\chi_{507}(223,\cdot)\)
\(\chi_{507}(232,\cdot)\)
\(\chi_{507}(241,\cdot)\)
\(\chi_{507}(253,\cdot)\)
\(\chi_{507}(262,\cdot)\)
\(\chi_{507}(271,\cdot)\)
\(\chi_{507}(280,\cdot)\)
\(\chi_{507}(292,\cdot)\)
\(\chi_{507}(301,\cdot)\)
...
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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 };
\((170,340)\) → \((1,e\left(\frac{1}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 507 }(340, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{156}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{3}{52}\right)\) | \(e\left(\frac{107}{156}\right)\) | \(e\left(\frac{1}{52}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{103}{156}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{73}{78}\right)\) |
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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)
