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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10304, base_ring=CyclotomicField(176))
M = H._module
chi = DirichletCharacter(H, M([0,77,88,152]))
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gp:[g,chi] = znchar(Mod(237, 10304))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("10304.237");
| Modulus: | \(10304\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(10304\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(176\) |
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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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\(\chi_{10304}(125,\cdot)\)
\(\chi_{10304}(181,\cdot)\)
\(\chi_{10304}(237,\cdot)\)
\(\chi_{10304}(293,\cdot)\)
\(\chi_{10304}(405,\cdot)\)
\(\chi_{10304}(517,\cdot)\)
\(\chi_{10304}(573,\cdot)\)
\(\chi_{10304}(741,\cdot)\)
\(\chi_{10304}(797,\cdot)\)
\(\chi_{10304}(1077,\cdot)\)
\(\chi_{10304}(1413,\cdot)\)
\(\chi_{10304}(1469,\cdot)\)
\(\chi_{10304}(1525,\cdot)\)
\(\chi_{10304}(1581,\cdot)\)
\(\chi_{10304}(1693,\cdot)\)
\(\chi_{10304}(1805,\cdot)\)
\(\chi_{10304}(1861,\cdot)\)
\(\chi_{10304}(2029,\cdot)\)
\(\chi_{10304}(2085,\cdot)\)
\(\chi_{10304}(2365,\cdot)\)
\(\chi_{10304}(2701,\cdot)\)
\(\chi_{10304}(2757,\cdot)\)
\(\chi_{10304}(2813,\cdot)\)
\(\chi_{10304}(2869,\cdot)\)
\(\chi_{10304}(2981,\cdot)\)
\(\chi_{10304}(3093,\cdot)\)
\(\chi_{10304}(3149,\cdot)\)
\(\chi_{10304}(3317,\cdot)\)
\(\chi_{10304}(3373,\cdot)\)
\(\chi_{10304}(3653,\cdot)\)
...
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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 };
| Field of values: |
$\Q(\zeta_{176})$ |
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sage:CyclotomicField(chi.multiplicative_order())
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gp:nfinit(polcyclo(charorder(g,chi)))
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magma:CyclotomicField(Order(chi));
|
| Fixed field: |
Number field defined by a degree 176 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((9983,645,1473,6721)\) → \((1,e\left(\frac{7}{16}\right),-1,e\left(\frac{19}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(25\) | \(27\) |
| \( \chi_{ 10304 }(237, a) \) |
\(1\) | \(1\) | \(e\left(\frac{111}{176}\right)\) | \(e\left(\frac{141}{176}\right)\) | \(e\left(\frac{23}{88}\right)\) | \(e\left(\frac{169}{176}\right)\) | \(e\left(\frac{27}{176}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{91}{176}\right)\) | \(e\left(\frac{53}{88}\right)\) | \(e\left(\frac{157}{176}\right)\) |
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sage:chi(x) # x integer
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gp:chareval(g,chi,x) \\ x integer, value in Q/Z
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magma:chi(x)
