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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(20736, base_ring=CyclotomicField(288))
M = H._module
chi = DirichletCharacter(H, M([144,261,16]))
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gp:[g,chi] = znchar(Mod(1223, 20736))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("20736.1223");
| Modulus: | \(20736\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3456\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(288\) |
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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 \(\chi_{3456}(299,\cdot)\) |
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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: | no |
| 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);
|
\(\chi_{20736}(71,\cdot)\)
\(\chi_{20736}(359,\cdot)\)
\(\chi_{20736}(503,\cdot)\)
\(\chi_{20736}(791,\cdot)\)
\(\chi_{20736}(935,\cdot)\)
\(\chi_{20736}(1223,\cdot)\)
\(\chi_{20736}(1367,\cdot)\)
\(\chi_{20736}(1655,\cdot)\)
\(\chi_{20736}(1799,\cdot)\)
\(\chi_{20736}(2087,\cdot)\)
\(\chi_{20736}(2231,\cdot)\)
\(\chi_{20736}(2519,\cdot)\)
\(\chi_{20736}(2663,\cdot)\)
\(\chi_{20736}(2951,\cdot)\)
\(\chi_{20736}(3095,\cdot)\)
\(\chi_{20736}(3383,\cdot)\)
\(\chi_{20736}(3527,\cdot)\)
\(\chi_{20736}(3815,\cdot)\)
\(\chi_{20736}(3959,\cdot)\)
\(\chi_{20736}(4247,\cdot)\)
\(\chi_{20736}(4391,\cdot)\)
\(\chi_{20736}(4679,\cdot)\)
\(\chi_{20736}(4823,\cdot)\)
\(\chi_{20736}(5111,\cdot)\)
\(\chi_{20736}(5255,\cdot)\)
\(\chi_{20736}(5543,\cdot)\)
\(\chi_{20736}(5687,\cdot)\)
\(\chi_{20736}(5975,\cdot)\)
\(\chi_{20736}(6119,\cdot)\)
\(\chi_{20736}(6407,\cdot)\)
...
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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 };
\((12799,15877,6401)\) → \((-1,e\left(\frac{29}{32}\right),e\left(\frac{1}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 20736 }(1223, a) \) |
\(1\) | \(1\) | \(e\left(\frac{53}{288}\right)\) | \(e\left(\frac{65}{144}\right)\) | \(e\left(\frac{73}{288}\right)\) | \(e\left(\frac{11}{288}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{1}{96}\right)\) | \(e\left(\frac{115}{144}\right)\) | \(e\left(\frac{53}{144}\right)\) | \(e\left(\frac{151}{288}\right)\) | \(e\left(\frac{31}{36}\right)\) |
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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)
