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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(40075, base_ring=CyclotomicField(114))
M = H._module
chi = DirichletCharacter(H, M([0,76,27]))
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gp:[g,chi] = znchar(Mod(24476, 40075))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("40075.24476");
| Modulus: | \(40075\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1603\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(114\) |
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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 \(\chi_{1603}(431,\cdot)\) |
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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: | 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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\(\chi_{40075}(401,\cdot)\)
\(\chi_{40075}(751,\cdot)\)
\(\chi_{40075}(1101,\cdot)\)
\(\chi_{40075}(2076,\cdot)\)
\(\chi_{40075}(4001,\cdot)\)
\(\chi_{40075}(5751,\cdot)\)
\(\chi_{40075}(6351,\cdot)\)
\(\chi_{40075}(8026,\cdot)\)
\(\chi_{40075}(8201,\cdot)\)
\(\chi_{40075}(9601,\cdot)\)
\(\chi_{40075}(9851,\cdot)\)
\(\chi_{40075}(10201,\cdot)\)
\(\chi_{40075}(13451,\cdot)\)
\(\chi_{40075}(13526,\cdot)\)
\(\chi_{40075}(15451,\cdot)\)
\(\chi_{40075}(17201,\cdot)\)
\(\chi_{40075}(19476,\cdot)\)
\(\chi_{40075}(19651,\cdot)\)
\(\chi_{40075}(21051,\cdot)\)
\(\chi_{40075}(23076,\cdot)\)
\(\chi_{40075}(23426,\cdot)\)
\(\chi_{40075}(24476,\cdot)\)
\(\chi_{40075}(24901,\cdot)\)
\(\chi_{40075}(26751,\cdot)\)
\(\chi_{40075}(28151,\cdot)\)
\(\chi_{40075}(29026,\cdot)\)
\(\chi_{40075}(29376,\cdot)\)
\(\chi_{40075}(29726,\cdot)\)
\(\chi_{40075}(34526,\cdot)\)
\(\chi_{40075}(34876,\cdot)\)
...
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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 };
\((27252,28626,12601)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{9}{38}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 40075 }(24476, a) \) |
\(1\) | \(1\) | \(e\left(\frac{35}{114}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{13}{57}\right)\) |
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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)
