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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4760, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,8,3]))
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gp:[g,chi] = znchar(Mod(3121, 4760))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4760.3121");
| Modulus: | \(4760\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(119\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(16\) |
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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 \(\chi_{119}(27,\cdot)\) |
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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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\(\chi_{4760}(41,\cdot)\)
\(\chi_{4760}(601,\cdot)\)
\(\chi_{4760}(881,\cdot)\)
\(\chi_{4760}(1161,\cdot)\)
\(\chi_{4760}(2001,\cdot)\)
\(\chi_{4760}(2281,\cdot)\)
\(\chi_{4760}(2561,\cdot)\)
\(\chi_{4760}(3121,\cdot)\)
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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 };
\((1191,2381,2857,1361,3641)\) → \((1,1,1,-1,e\left(\frac{3}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(33\) |
| \( \chi_{ 4760 }(3121, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(i\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(1\) |
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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)
