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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(24200, base_ring=CyclotomicField(220))
M = H._module
chi = DirichletCharacter(H, M([110,110,209,20]))
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gp:[g,chi] = znchar(Mod(8163, 24200))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("24200.8163");
| Modulus: | \(24200\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(24200\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(220\) |
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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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\(\chi_{24200}(67,\cdot)\)
\(\chi_{24200}(683,\cdot)\)
\(\chi_{24200}(947,\cdot)\)
\(\chi_{24200}(1123,\cdot)\)
\(\chi_{24200}(1387,\cdot)\)
\(\chi_{24200}(1563,\cdot)\)
\(\chi_{24200}(1827,\cdot)\)
\(\chi_{24200}(2003,\cdot)\)
\(\chi_{24200}(2267,\cdot)\)
\(\chi_{24200}(2883,\cdot)\)
\(\chi_{24200}(3323,\cdot)\)
\(\chi_{24200}(3587,\cdot)\)
\(\chi_{24200}(3763,\cdot)\)
\(\chi_{24200}(4027,\cdot)\)
\(\chi_{24200}(4203,\cdot)\)
\(\chi_{24200}(4467,\cdot)\)
\(\chi_{24200}(5347,\cdot)\)
\(\chi_{24200}(5523,\cdot)\)
\(\chi_{24200}(5787,\cdot)\)
\(\chi_{24200}(5963,\cdot)\)
\(\chi_{24200}(6227,\cdot)\)
\(\chi_{24200}(6403,\cdot)\)
\(\chi_{24200}(6667,\cdot)\)
\(\chi_{24200}(7283,\cdot)\)
\(\chi_{24200}(7547,\cdot)\)
\(\chi_{24200}(7723,\cdot)\)
\(\chi_{24200}(8163,\cdot)\)
\(\chi_{24200}(8427,\cdot)\)
\(\chi_{24200}(8603,\cdot)\)
\(\chi_{24200}(8867,\cdot)\)
...
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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: |
$\Q(\zeta_{220})$ |
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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 220 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((18151,12101,6777,14401)\) → \((-1,-1,e\left(\frac{19}{20}\right),e\left(\frac{1}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 24200 }(8163, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{161}{220}\right)\) | \(e\left(\frac{177}{220}\right)\) | \(e\left(\frac{71}{110}\right)\) | \(e\left(\frac{59}{110}\right)\) | \(e\left(\frac{69}{220}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{52}{55}\right)\) |
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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)
