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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1224, base_ring=CyclotomicField(6))
M = H._module
chi = DirichletCharacter(H, M([3,3,5,3]))
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gp:[g,chi] = znchar(Mod(203, 1224))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1224.203");
| Modulus: | \(1224\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1224\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(6\) |
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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: | 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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\(\chi_{1224}(203,\cdot)\)
\(\chi_{1224}(1019,\cdot)\)
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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 };
\((919,613,137,649)\) → \((-1,-1,e\left(\frac{5}{6}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 1224 }(203, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) |
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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)
