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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(24336, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([78,39,78,92]))
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gp:[g,chi] = znchar(Mod(8459, 24336))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("24336.8459");
| Modulus: | \(24336\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(8112\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(156\) |
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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 \(\chi_{8112}(347,\cdot)\) |
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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);
|
\(\chi_{24336}(35,\cdot)\)
\(\chi_{24336}(107,\cdot)\)
\(\chi_{24336}(971,\cdot)\)
\(\chi_{24336}(1043,\cdot)\)
\(\chi_{24336}(1907,\cdot)\)
\(\chi_{24336}(1979,\cdot)\)
\(\chi_{24336}(2843,\cdot)\)
\(\chi_{24336}(2915,\cdot)\)
\(\chi_{24336}(3779,\cdot)\)
\(\chi_{24336}(3851,\cdot)\)
\(\chi_{24336}(4715,\cdot)\)
\(\chi_{24336}(4787,\cdot)\)
\(\chi_{24336}(5651,\cdot)\)
\(\chi_{24336}(6587,\cdot)\)
\(\chi_{24336}(6659,\cdot)\)
\(\chi_{24336}(7523,\cdot)\)
\(\chi_{24336}(7595,\cdot)\)
\(\chi_{24336}(8459,\cdot)\)
\(\chi_{24336}(8531,\cdot)\)
\(\chi_{24336}(9395,\cdot)\)
\(\chi_{24336}(9467,\cdot)\)
\(\chi_{24336}(10403,\cdot)\)
\(\chi_{24336}(11267,\cdot)\)
\(\chi_{24336}(11339,\cdot)\)
\(\chi_{24336}(12203,\cdot)\)
\(\chi_{24336}(12275,\cdot)\)
\(\chi_{24336}(13139,\cdot)\)
\(\chi_{24336}(13211,\cdot)\)
\(\chi_{24336}(14075,\cdot)\)
\(\chi_{24336}(14147,\cdot)\)
...
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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 };
\((21295,6085,18929,3889)\) → \((-1,i,-1,e\left(\frac{23}{39}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 24336 }(8459, a) \) |
\(1\) | \(1\) | \(e\left(\frac{3}{52}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{155}{156}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{131}{156}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{25}{156}\right)\) |
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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)
