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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6003, base_ring=CyclotomicField(132))
M = H._module
chi = DirichletCharacter(H, M([110,120,99]))
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gp:[g,chi] = znchar(Mod(104, 6003))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6003.104");
| Modulus: | \(6003\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(6003\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(132\) |
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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);
|
\(\chi_{6003}(41,\cdot)\)
\(\chi_{6003}(104,\cdot)\)
\(\chi_{6003}(128,\cdot)\)
\(\chi_{6003}(302,\cdot)\)
\(\chi_{6003}(650,\cdot)\)
\(\chi_{6003}(887,\cdot)\)
\(\chi_{6003}(974,\cdot)\)
\(\chi_{6003}(1085,\cdot)\)
\(\chi_{6003}(1235,\cdot)\)
\(\chi_{6003}(1346,\cdot)\)
\(\chi_{6003}(1409,\cdot)\)
\(\chi_{6003}(1757,\cdot)\)
\(\chi_{6003}(2129,\cdot)\)
\(\chi_{6003}(2216,\cdot)\)
\(\chi_{6003}(2279,\cdot)\)
\(\chi_{6003}(2477,\cdot)\)
\(\chi_{6003}(2651,\cdot)\)
\(\chi_{6003}(2801,\cdot)\)
\(\chi_{6003}(2975,\cdot)\)
\(\chi_{6003}(2999,\cdot)\)
\(\chi_{6003}(3062,\cdot)\)
\(\chi_{6003}(3236,\cdot)\)
\(\chi_{6003}(3521,\cdot)\)
\(\chi_{6003}(3758,\cdot)\)
\(\chi_{6003}(3845,\cdot)\)
\(\chi_{6003}(4043,\cdot)\)
\(\chi_{6003}(4106,\cdot)\)
\(\chi_{6003}(4217,\cdot)\)
\(\chi_{6003}(4280,\cdot)\)
\(\chi_{6003}(4304,\cdot)\)
...
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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 };
\((668,3133,4555)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{10}{11}\right),-i)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 6003 }(104, a) \) |
\(1\) | \(1\) | \(e\left(\frac{53}{132}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{101}{132}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{1}{132}\right)\) | \(e\left(\frac{20}{33}\right)\) |
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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)
