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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5767, base_ring=CyclotomicField(936))
M = H._module
chi = DirichletCharacter(H, M([169,756]))
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gp:[g,chi] = znchar(Mod(410, 5767))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5767.410");
| Modulus: | \(5767\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5767\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(936\) |
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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);
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\(\chi_{5767}(14,\cdot)\)
\(\chi_{5767}(15,\cdot)\)
\(\chi_{5767}(33,\cdot)\)
\(\chi_{5767}(58,\cdot)\)
\(\chi_{5767}(93,\cdot)\)
\(\chi_{5767}(106,\cdot)\)
\(\chi_{5767}(112,\cdot)\)
\(\chi_{5767}(120,\cdot)\)
\(\chi_{5767}(172,\cdot)\)
\(\chi_{5767}(175,\cdot)\)
\(\chi_{5767}(185,\cdot)\)
\(\chi_{5767}(191,\cdot)\)
\(\chi_{5767}(199,\cdot)\)
\(\chi_{5767}(252,\cdot)\)
\(\chi_{5767}(264,\cdot)\)
\(\chi_{5767}(278,\cdot)\)
\(\chi_{5767}(306,\cdot)\)
\(\chi_{5767}(331,\cdot)\)
\(\chi_{5767}(385,\cdot)\)
\(\chi_{5767}(407,\cdot)\)
\(\chi_{5767}(409,\cdot)\)
\(\chi_{5767}(410,\cdot)\)
\(\chi_{5767}(412,\cdot)\)
\(\chi_{5767}(452,\cdot)\)
\(\chi_{5767}(453,\cdot)\)
\(\chi_{5767}(464,\cdot)\)
\(\chi_{5767}(466,\cdot)\)
\(\chi_{5767}(491,\cdot)\)
\(\chi_{5767}(531,\cdot)\)
\(\chi_{5767}(545,\cdot)\)
...
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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 };
\((1976,1899)\) → \((e\left(\frac{13}{72}\right),e\left(\frac{21}{26}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 5767 }(410, a) \) |
\(1\) | \(1\) | \(e\left(\frac{79}{117}\right)\) | \(e\left(\frac{139}{156}\right)\) | \(e\left(\frac{41}{117}\right)\) | \(e\left(\frac{241}{936}\right)\) | \(e\left(\frac{265}{468}\right)\) | \(e\left(\frac{239}{312}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{97}{104}\right)\) | \(e\left(\frac{799}{936}\right)\) |
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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)
