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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6253, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([31,117]))
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gp:[g,chi] = znchar(Mod(3521, 6253))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6253.3521");
| Modulus: | \(6253\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(6253\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(156\) |
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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;
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| 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);
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| 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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\(\chi_{6253}(154,\cdot)\)
\(\chi_{6253}(228,\cdot)\)
\(\chi_{6253}(253,\cdot)\)
\(\chi_{6253}(327,\cdot)\)
\(\chi_{6253}(635,\cdot)\)
\(\chi_{6253}(709,\cdot)\)
\(\chi_{6253}(734,\cdot)\)
\(\chi_{6253}(808,\cdot)\)
\(\chi_{6253}(1116,\cdot)\)
\(\chi_{6253}(1190,\cdot)\)
\(\chi_{6253}(1215,\cdot)\)
\(\chi_{6253}(1289,\cdot)\)
\(\chi_{6253}(1597,\cdot)\)
\(\chi_{6253}(1696,\cdot)\)
\(\chi_{6253}(2078,\cdot)\)
\(\chi_{6253}(2152,\cdot)\)
\(\chi_{6253}(2177,\cdot)\)
\(\chi_{6253}(2251,\cdot)\)
\(\chi_{6253}(2559,\cdot)\)
\(\chi_{6253}(2633,\cdot)\)
\(\chi_{6253}(2658,\cdot)\)
\(\chi_{6253}(2732,\cdot)\)
\(\chi_{6253}(3040,\cdot)\)
\(\chi_{6253}(3114,\cdot)\)
\(\chi_{6253}(3139,\cdot)\)
\(\chi_{6253}(3213,\cdot)\)
\(\chi_{6253}(3521,\cdot)\)
\(\chi_{6253}(3595,\cdot)\)
\(\chi_{6253}(3620,\cdot)\)
\(\chi_{6253}(3694,\cdot)\)
...
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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: |
$\Q(\zeta_{156})$ |
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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));
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| Fixed field: |
Number field defined by a degree 156 polynomial (not computed) |
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sage:chi.fixed_field()
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\((1185,5071)\) → \((e\left(\frac{31}{156}\right),-i)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 6253 }(3521, a) \) |
\(1\) | \(1\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{41}{156}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{151}{156}\right)\) |
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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)
