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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(15575, base_ring=CyclotomicField(88))
M = H._module
chi = DirichletCharacter(H, M([66,44,71]))
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gp:[g,chi] = znchar(Mod(11318, 15575))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("15575.11318");
| Modulus: | \(15575\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3115\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(88\) |
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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 \(\chi_{3115}(1973,\cdot)\) |
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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: | odd |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{15575}(118,\cdot)\)
\(\chi_{15575}(132,\cdot)\)
\(\chi_{15575}(293,\cdot)\)
\(\chi_{15575}(1007,\cdot)\)
\(\chi_{15575}(2232,\cdot)\)
\(\chi_{15575}(2568,\cdot)\)
\(\chi_{15575}(2918,\cdot)\)
\(\chi_{15575}(3457,\cdot)\)
\(\chi_{15575}(3618,\cdot)\)
\(\chi_{15575}(3982,\cdot)\)
\(\chi_{15575}(5032,\cdot)\)
\(\chi_{15575}(5193,\cdot)\)
\(\chi_{15575}(5893,\cdot)\)
\(\chi_{15575}(5907,\cdot)\)
\(\chi_{15575}(6243,\cdot)\)
\(\chi_{15575}(6432,\cdot)\)
\(\chi_{15575}(7482,\cdot)\)
\(\chi_{15575}(7657,\cdot)\)
\(\chi_{15575}(8007,\cdot)\)
\(\chi_{15575}(8182,\cdot)\)
\(\chi_{15575}(8518,\cdot)\)
\(\chi_{15575}(8693,\cdot)\)
\(\chi_{15575}(9043,\cdot)\)
\(\chi_{15575}(9218,\cdot)\)
\(\chi_{15575}(9232,\cdot)\)
\(\chi_{15575}(9757,\cdot)\)
\(\chi_{15575}(10443,\cdot)\)
\(\chi_{15575}(10618,\cdot)\)
\(\chi_{15575}(10632,\cdot)\)
\(\chi_{15575}(11318,\cdot)\)
...
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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 };
\((7477,11126,5076)\) → \((-i,-1,e\left(\frac{71}{88}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 15575 }(11318, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{49}{88}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{19}{88}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{27}{88}\right)\) | \(e\left(\frac{7}{11}\right)\) |
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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)
