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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1832, base_ring=CyclotomicField(114))
M = H._module
chi = DirichletCharacter(H, M([57,57,49]))
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gp:[g,chi] = znchar(Mod(1379, 1832))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1832.1379");
| Modulus: | \(1832\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1832\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(114\) |
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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: | yes |
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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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\(\chi_{1832}(99,\cdot)\)
\(\chi_{1832}(147,\cdot)\)
\(\chi_{1832}(275,\cdot)\)
\(\chi_{1832}(291,\cdot)\)
\(\chi_{1832}(299,\cdot)\)
\(\chi_{1832}(307,\cdot)\)
\(\chi_{1832}(347,\cdot)\)
\(\chi_{1832}(403,\cdot)\)
\(\chi_{1832}(491,\cdot)\)
\(\chi_{1832}(507,\cdot)\)
\(\chi_{1832}(555,\cdot)\)
\(\chi_{1832}(667,\cdot)\)
\(\chi_{1832}(699,\cdot)\)
\(\chi_{1832}(723,\cdot)\)
\(\chi_{1832}(763,\cdot)\)
\(\chi_{1832}(787,\cdot)\)
\(\chi_{1832}(835,\cdot)\)
\(\chi_{1832}(891,\cdot)\)
\(\chi_{1832}(907,\cdot)\)
\(\chi_{1832}(987,\cdot)\)
\(\chi_{1832}(1019,\cdot)\)
\(\chi_{1832}(1131,\cdot)\)
\(\chi_{1832}(1203,\cdot)\)
\(\chi_{1832}(1283,\cdot)\)
\(\chi_{1832}(1291,\cdot)\)
\(\chi_{1832}(1299,\cdot)\)
\(\chi_{1832}(1323,\cdot)\)
\(\chi_{1832}(1355,\cdot)\)
\(\chi_{1832}(1371,\cdot)\)
\(\chi_{1832}(1379,\cdot)\)
...
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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 };
\((1375,917,1609)\) → \((-1,-1,e\left(\frac{49}{114}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1832 }(1379, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{71}{114}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{17}{19}\right)\) |
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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)
