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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(89712, base_ring=CyclotomicField(264))
M = H._module
chi = DirichletCharacter(H, M([0,66,44,0,87]))
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gp:[g,chi] = znchar(Mod(23717, 89712))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("89712.23717");
| Modulus: | \(89712\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(12816\) |
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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: | no, induced from \(\chi_{12816}(10901,\cdot)\) |
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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: | 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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\(\chi_{89712}(29,\cdot)\)
\(\chi_{89712}(1037,\cdot)\)
\(\chi_{89712}(1373,\cdot)\)
\(\chi_{89712}(1541,\cdot)\)
\(\chi_{89712}(3053,\cdot)\)
\(\chi_{89712}(3557,\cdot)\)
\(\chi_{89712}(3893,\cdot)\)
\(\chi_{89712}(4901,\cdot)\)
\(\chi_{89712}(7085,\cdot)\)
\(\chi_{89712}(7589,\cdot)\)
\(\chi_{89712}(8429,\cdot)\)
\(\chi_{89712}(8933,\cdot)\)
\(\chi_{89712}(9605,\cdot)\)
\(\chi_{89712}(13469,\cdot)\)
\(\chi_{89712}(14477,\cdot)\)
\(\chi_{89712}(15149,\cdot)\)
\(\chi_{89712}(19013,\cdot)\)
\(\chi_{89712}(19517,\cdot)\)
\(\chi_{89712}(21197,\cdot)\)
\(\chi_{89712}(22709,\cdot)\)
\(\chi_{89712}(23045,\cdot)\)
\(\chi_{89712}(23717,\cdot)\)
\(\chi_{89712}(24053,\cdot)\)
\(\chi_{89712}(24221,\cdot)\)
\(\chi_{89712}(26573,\cdot)\)
\(\chi_{89712}(26741,\cdot)\)
\(\chi_{89712}(31109,\cdot)\)
\(\chi_{89712}(31277,\cdot)\)
\(\chi_{89712}(33629,\cdot)\)
\(\chi_{89712}(33797,\cdot)\)
...
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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 };
\((11215,67285,19937,64081,86689)\) → \((1,i,e\left(\frac{1}{6}\right),1,e\left(\frac{29}{88}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 89712 }(23717, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{13}{132}\right)\) | \(e\left(\frac{175}{264}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{25}{88}\right)\) | \(e\left(\frac{31}{264}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{95}{264}\right)\) | \(e\left(\frac{145}{264}\right)\) | \(e\left(\frac{7}{8}\right)\) |
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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)
