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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(369, base_ring=CyclotomicField(120))
M = H._module
chi = DirichletCharacter(H, M([100,111]))
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gp:[g,chi] = znchar(Mod(302, 369))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("369.302");
| Modulus: | \(369\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(369\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(120\) |
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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);
|
\(\chi_{369}(11,\cdot)\)
\(\chi_{369}(29,\cdot)\)
\(\chi_{369}(47,\cdot)\)
\(\chi_{369}(56,\cdot)\)
\(\chi_{369}(65,\cdot)\)
\(\chi_{369}(95,\cdot)\)
\(\chi_{369}(101,\cdot)\)
\(\chi_{369}(104,\cdot)\)
\(\chi_{369}(110,\cdot)\)
\(\chi_{369}(140,\cdot)\)
\(\chi_{369}(149,\cdot)\)
\(\chi_{369}(158,\cdot)\)
\(\chi_{369}(176,\cdot)\)
\(\chi_{369}(194,\cdot)\)
\(\chi_{369}(212,\cdot)\)
\(\chi_{369}(218,\cdot)\)
\(\chi_{369}(227,\cdot)\)
\(\chi_{369}(239,\cdot)\)
\(\chi_{369}(257,\cdot)\)
\(\chi_{369}(263,\cdot)\)
\(\chi_{369}(272,\cdot)\)
\(\chi_{369}(275,\cdot)\)
\(\chi_{369}(281,\cdot)\)
\(\chi_{369}(293,\cdot)\)
\(\chi_{369}(299,\cdot)\)
\(\chi_{369}(302,\cdot)\)
\(\chi_{369}(311,\cdot)\)
\(\chi_{369}(317,\cdot)\)
\(\chi_{369}(335,\cdot)\)
\(\chi_{369}(347,\cdot)\)
...
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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 };
\((83,334)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{37}{40}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 369 }(302, a) \) |
\(1\) | \(1\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{49}{120}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{41}{120}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{8}{15}\right)\) |
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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)
