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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(747, base_ring=CyclotomicField(82))
M = H._module
chi = DirichletCharacter(H, M([41,8]))
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gp:[g,chi] = znchar(Mod(422, 747))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("747.422");
| Modulus: | \(747\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(249\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(82\) |
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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 \(\chi_{249}(173,\cdot)\) |
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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);
|
\(\chi_{747}(17,\cdot)\)
\(\chi_{747}(26,\cdot)\)
\(\chi_{747}(44,\cdot)\)
\(\chi_{747}(116,\cdot)\)
\(\chi_{747}(134,\cdot)\)
\(\chi_{747}(152,\cdot)\)
\(\chi_{747}(161,\cdot)\)
\(\chi_{747}(170,\cdot)\)
\(\chi_{747}(197,\cdot)\)
\(\chi_{747}(206,\cdot)\)
\(\chi_{747}(215,\cdot)\)
\(\chi_{747}(260,\cdot)\)
\(\chi_{747}(278,\cdot)\)
\(\chi_{747}(287,\cdot)\)
\(\chi_{747}(314,\cdot)\)
\(\chi_{747}(341,\cdot)\)
\(\chi_{747}(359,\cdot)\)
\(\chi_{747}(368,\cdot)\)
\(\chi_{747}(395,\cdot)\)
\(\chi_{747}(413,\cdot)\)
\(\chi_{747}(422,\cdot)\)
\(\chi_{747}(431,\cdot)\)
\(\chi_{747}(440,\cdot)\)
\(\chi_{747}(476,\cdot)\)
\(\chi_{747}(485,\cdot)\)
\(\chi_{747}(521,\cdot)\)
\(\chi_{747}(539,\cdot)\)
\(\chi_{747}(557,\cdot)\)
\(\chi_{747}(566,\cdot)\)
\(\chi_{747}(575,\cdot)\)
...
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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 };
\((416,334)\) → \((-1,e\left(\frac{4}{41}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 747 }(422, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{49}{82}\right)\) | \(e\left(\frac{8}{41}\right)\) | \(e\left(\frac{11}{82}\right)\) | \(e\left(\frac{32}{41}\right)\) | \(e\left(\frac{65}{82}\right)\) | \(e\left(\frac{30}{41}\right)\) | \(e\left(\frac{69}{82}\right)\) | \(e\left(\frac{21}{41}\right)\) | \(e\left(\frac{31}{82}\right)\) | \(e\left(\frac{16}{41}\right)\) |
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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)
