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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(14784, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([40,65,0,40,56]))
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gp:[g,chi] = znchar(Mod(7531, 14784))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("14784.7531");
| Modulus: | \(14784\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4928\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(80\) |
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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 \(\chi_{4928}(2603,\cdot)\) |
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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);
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\(\chi_{14784}(139,\cdot)\)
\(\chi_{14784}(475,\cdot)\)
\(\chi_{14784}(811,\cdot)\)
\(\chi_{14784}(1315,\cdot)\)
\(\chi_{14784}(1987,\cdot)\)
\(\chi_{14784}(2323,\cdot)\)
\(\chi_{14784}(2659,\cdot)\)
\(\chi_{14784}(3163,\cdot)\)
\(\chi_{14784}(3835,\cdot)\)
\(\chi_{14784}(4171,\cdot)\)
\(\chi_{14784}(4507,\cdot)\)
\(\chi_{14784}(5011,\cdot)\)
\(\chi_{14784}(5683,\cdot)\)
\(\chi_{14784}(6019,\cdot)\)
\(\chi_{14784}(6355,\cdot)\)
\(\chi_{14784}(6859,\cdot)\)
\(\chi_{14784}(7531,\cdot)\)
\(\chi_{14784}(7867,\cdot)\)
\(\chi_{14784}(8203,\cdot)\)
\(\chi_{14784}(8707,\cdot)\)
\(\chi_{14784}(9379,\cdot)\)
\(\chi_{14784}(9715,\cdot)\)
\(\chi_{14784}(10051,\cdot)\)
\(\chi_{14784}(10555,\cdot)\)
\(\chi_{14784}(11227,\cdot)\)
\(\chi_{14784}(11563,\cdot)\)
\(\chi_{14784}(11899,\cdot)\)
\(\chi_{14784}(12403,\cdot)\)
\(\chi_{14784}(13075,\cdot)\)
\(\chi_{14784}(13411,\cdot)\)
...
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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 };
\((4159,6469,9857,12673,8065)\) → \((-1,e\left(\frac{13}{16}\right),1,-1,e\left(\frac{7}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 14784 }(7531, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{39}{40}\right)\) |
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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)
