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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4869, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([90,121]))
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gp:[g,chi] = znchar(Mod(1124, 4869))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4869.1124");
| Modulus: | \(4869\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1623\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(180\) |
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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 \(\chi_{1623}(1124,\cdot)\) |
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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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\(\chi_{4869}(8,\cdot)\)
\(\chi_{4869}(71,\cdot)\)
\(\chi_{4869}(242,\cdot)\)
\(\chi_{4869}(305,\cdot)\)
\(\chi_{4869}(422,\cdot)\)
\(\chi_{4869}(503,\cdot)\)
\(\chi_{4869}(647,\cdot)\)
\(\chi_{4869}(701,\cdot)\)
\(\chi_{4869}(827,\cdot)\)
\(\chi_{4869}(863,\cdot)\)
\(\chi_{4869}(881,\cdot)\)
\(\chi_{4869}(971,\cdot)\)
\(\chi_{4869}(1043,\cdot)\)
\(\chi_{4869}(1115,\cdot)\)
\(\chi_{4869}(1124,\cdot)\)
\(\chi_{4869}(1259,\cdot)\)
\(\chi_{4869}(1466,\cdot)\)
\(\chi_{4869}(1475,\cdot)\)
\(\chi_{4869}(1673,\cdot)\)
\(\chi_{4869}(1772,\cdot)\)
\(\chi_{4869}(1826,\cdot)\)
\(\chi_{4869}(1961,\cdot)\)
\(\chi_{4869}(2015,\cdot)\)
\(\chi_{4869}(2114,\cdot)\)
\(\chi_{4869}(2312,\cdot)\)
\(\chi_{4869}(2321,\cdot)\)
\(\chi_{4869}(2528,\cdot)\)
\(\chi_{4869}(2663,\cdot)\)
\(\chi_{4869}(2672,\cdot)\)
\(\chi_{4869}(2744,\cdot)\)
...
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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 };
\((542,4330)\) → \((-1,e\left(\frac{121}{180}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 4869 }(1124, a) \) |
\(1\) | \(1\) | \(e\left(\frac{31}{180}\right)\) | \(e\left(\frac{31}{90}\right)\) | \(e\left(\frac{83}{90}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{17}{180}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{73}{180}\right)\) | \(e\left(\frac{109}{180}\right)\) | \(e\left(\frac{31}{45}\right)\) |
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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)
