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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(675, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([100,153]))
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gp:[g,chi] = znchar(Mod(322, 675))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("675.322");
| Modulus: | \(675\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(675\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(180\) |
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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);
|
\(\chi_{675}(13,\cdot)\)
\(\chi_{675}(22,\cdot)\)
\(\chi_{675}(52,\cdot)\)
\(\chi_{675}(58,\cdot)\)
\(\chi_{675}(67,\cdot)\)
\(\chi_{675}(88,\cdot)\)
\(\chi_{675}(97,\cdot)\)
\(\chi_{675}(103,\cdot)\)
\(\chi_{675}(112,\cdot)\)
\(\chi_{675}(133,\cdot)\)
\(\chi_{675}(142,\cdot)\)
\(\chi_{675}(148,\cdot)\)
\(\chi_{675}(178,\cdot)\)
\(\chi_{675}(187,\cdot)\)
\(\chi_{675}(202,\cdot)\)
\(\chi_{675}(223,\cdot)\)
\(\chi_{675}(238,\cdot)\)
\(\chi_{675}(247,\cdot)\)
\(\chi_{675}(277,\cdot)\)
\(\chi_{675}(283,\cdot)\)
\(\chi_{675}(292,\cdot)\)
\(\chi_{675}(313,\cdot)\)
\(\chi_{675}(322,\cdot)\)
\(\chi_{675}(328,\cdot)\)
\(\chi_{675}(337,\cdot)\)
\(\chi_{675}(358,\cdot)\)
\(\chi_{675}(367,\cdot)\)
\(\chi_{675}(373,\cdot)\)
\(\chi_{675}(403,\cdot)\)
\(\chi_{675}(412,\cdot)\)
...
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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 };
\((326,352)\) → \((e\left(\frac{5}{9}\right),e\left(\frac{17}{20}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 675 }(322, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{73}{180}\right)\) | \(e\left(\frac{73}{90}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{107}{180}\right)\) | \(e\left(\frac{49}{90}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) |
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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)
