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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(38112, base_ring=CyclotomicField(264))
M = H._module
chi = DirichletCharacter(H, M([132,231,132,62]))
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gp:[g,chi] = znchar(Mod(11219, 38112))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("38112.11219");
| Modulus: | \(38112\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(38112\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(264\) |
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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: | 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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\(\chi_{38112}(467,\cdot)\)
\(\chi_{38112}(563,\cdot)\)
\(\chi_{38112}(1259,\cdot)\)
\(\chi_{38112}(1403,\cdot)\)
\(\chi_{38112}(1475,\cdot)\)
\(\chi_{38112}(1931,\cdot)\)
\(\chi_{38112}(2435,\cdot)\)
\(\chi_{38112}(2507,\cdot)\)
\(\chi_{38112}(6227,\cdot)\)
\(\chi_{38112}(6299,\cdot)\)
\(\chi_{38112}(6803,\cdot)\)
\(\chi_{38112}(7259,\cdot)\)
\(\chi_{38112}(7331,\cdot)\)
\(\chi_{38112}(7475,\cdot)\)
\(\chi_{38112}(8171,\cdot)\)
\(\chi_{38112}(8267,\cdot)\)
\(\chi_{38112}(8915,\cdot)\)
\(\chi_{38112}(10307,\cdot)\)
\(\chi_{38112}(11099,\cdot)\)
\(\chi_{38112}(11171,\cdot)\)
\(\chi_{38112}(11219,\cdot)\)
\(\chi_{38112}(12587,\cdot)\)
\(\chi_{38112}(12827,\cdot)\)
\(\chi_{38112}(12971,\cdot)\)
\(\chi_{38112}(13187,\cdot)\)
\(\chi_{38112}(13403,\cdot)\)
\(\chi_{38112}(13427,\cdot)\)
\(\chi_{38112}(13835,\cdot)\)
\(\chi_{38112}(13955,\cdot)\)
\(\chi_{38112}(14363,\cdot)\)
...
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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: |
$\Q(\zeta_{264})$ |
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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 264 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((21439,33349,25409,17473)\) → \((-1,e\left(\frac{7}{8}\right),-1,e\left(\frac{31}{132}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 38112 }(11219, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{161}{264}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{67}{264}\right)\) | \(e\left(\frac{223}{264}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{181}{264}\right)\) | \(e\left(\frac{83}{132}\right)\) | \(e\left(\frac{29}{132}\right)\) | \(e\left(\frac{73}{264}\right)\) | \(e\left(\frac{13}{22}\right)\) |
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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)
