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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6012, base_ring=CyclotomicField(166))
M = H._module
chi = DirichletCharacter(H, M([83,83,10]))
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gp:[g,chi] = znchar(Mod(467, 6012))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6012.467");
| Modulus: | \(6012\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2004\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(166\) |
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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 \(\chi_{2004}(467,\cdot)\) |
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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);
|
\(\chi_{6012}(107,\cdot)\)
\(\chi_{6012}(179,\cdot)\)
\(\chi_{6012}(215,\cdot)\)
\(\chi_{6012}(251,\cdot)\)
\(\chi_{6012}(359,\cdot)\)
\(\chi_{6012}(395,\cdot)\)
\(\chi_{6012}(431,\cdot)\)
\(\chi_{6012}(467,\cdot)\)
\(\chi_{6012}(503,\cdot)\)
\(\chi_{6012}(539,\cdot)\)
\(\chi_{6012}(755,\cdot)\)
\(\chi_{6012}(863,\cdot)\)
\(\chi_{6012}(899,\cdot)\)
\(\chi_{6012}(935,\cdot)\)
\(\chi_{6012}(1079,\cdot)\)
\(\chi_{6012}(1187,\cdot)\)
\(\chi_{6012}(1223,\cdot)\)
\(\chi_{6012}(1295,\cdot)\)
\(\chi_{6012}(1331,\cdot)\)
\(\chi_{6012}(1367,\cdot)\)
\(\chi_{6012}(1511,\cdot)\)
\(\chi_{6012}(1547,\cdot)\)
\(\chi_{6012}(1619,\cdot)\)
\(\chi_{6012}(1655,\cdot)\)
\(\chi_{6012}(1691,\cdot)\)
\(\chi_{6012}(1727,\cdot)\)
\(\chi_{6012}(1763,\cdot)\)
\(\chi_{6012}(2015,\cdot)\)
\(\chi_{6012}(2051,\cdot)\)
\(\chi_{6012}(2195,\cdot)\)
...
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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 };
\((3007,3341,4681)\) → \((-1,-1,e\left(\frac{5}{83}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 6012 }(467, a) \) |
\(1\) | \(1\) | \(e\left(\frac{93}{166}\right)\) | \(e\left(\frac{101}{166}\right)\) | \(e\left(\frac{57}{83}\right)\) | \(e\left(\frac{17}{83}\right)\) | \(e\left(\frac{115}{166}\right)\) | \(e\left(\frac{165}{166}\right)\) | \(e\left(\frac{80}{83}\right)\) | \(e\left(\frac{10}{83}\right)\) | \(e\left(\frac{89}{166}\right)\) | \(e\left(\frac{153}{166}\right)\) |
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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)
