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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(82365, base_ring=CyclotomicField(272))
M = H._module
chi = DirichletCharacter(H, M([0,68,61,136]))
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gp:[g,chi] = znchar(Mod(5452, 82365))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("82365.5452");
| Modulus: | \(82365\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(27455\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(272\) |
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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 \(\chi_{27455}(5452,\cdot)\) |
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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);
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\(\chi_{82365}(37,\cdot)\)
\(\chi_{82365}(607,\cdot)\)
\(\chi_{82365}(2203,\cdot)\)
\(\chi_{82365}(2317,\cdot)\)
\(\chi_{82365}(2488,\cdot)\)
\(\chi_{82365}(2887,\cdot)\)
\(\chi_{82365}(3343,\cdot)\)
\(\chi_{82365}(3628,\cdot)\)
\(\chi_{82365}(4882,\cdot)\)
\(\chi_{82365}(5452,\cdot)\)
\(\chi_{82365}(7048,\cdot)\)
\(\chi_{82365}(7162,\cdot)\)
\(\chi_{82365}(7333,\cdot)\)
\(\chi_{82365}(7732,\cdot)\)
\(\chi_{82365}(8188,\cdot)\)
\(\chi_{82365}(8473,\cdot)\)
\(\chi_{82365}(9727,\cdot)\)
\(\chi_{82365}(10297,\cdot)\)
\(\chi_{82365}(11893,\cdot)\)
\(\chi_{82365}(12577,\cdot)\)
\(\chi_{82365}(13033,\cdot)\)
\(\chi_{82365}(13318,\cdot)\)
\(\chi_{82365}(14572,\cdot)\)
\(\chi_{82365}(15142,\cdot)\)
\(\chi_{82365}(16738,\cdot)\)
\(\chi_{82365}(16852,\cdot)\)
\(\chi_{82365}(17023,\cdot)\)
\(\chi_{82365}(17422,\cdot)\)
\(\chi_{82365}(18163,\cdot)\)
\(\chi_{82365}(19417,\cdot)\)
...
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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 };
\((54911,32947,41041,56356)\) → \((1,i,e\left(\frac{61}{272}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(22\) | \(23\) |
| \( \chi_{ 82365 }(5452, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{49}{136}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{3}{272}\right)\) | \(e\left(\frac{11}{136}\right)\) | \(e\left(\frac{43}{272}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{101}{272}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{141}{272}\right)\) | \(e\left(\frac{207}{272}\right)\) |
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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)
