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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(16055, base_ring=CyclotomicField(468))
M = H._module
chi = DirichletCharacter(H, M([351,447,286]))
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gp:[g,chi] = znchar(Mod(2333, 16055))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("16055.2333");
| Modulus: | \(16055\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(16055\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(468\) |
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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: | odd |
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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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\(\chi_{16055}(72,\cdot)\)
\(\chi_{16055}(223,\cdot)\)
\(\chi_{16055}(288,\cdot)\)
\(\chi_{16055}(622,\cdot)\)
\(\chi_{16055}(687,\cdot)\)
\(\chi_{16055}(743,\cdot)\)
\(\chi_{16055}(773,\cdot)\)
\(\chi_{16055}(903,\cdot)\)
\(\chi_{16055}(982,\cdot)\)
\(\chi_{16055}(1098,\cdot)\)
\(\chi_{16055}(1112,\cdot)\)
\(\chi_{16055}(1142,\cdot)\)
\(\chi_{16055}(1307,\cdot)\)
\(\chi_{16055}(1458,\cdot)\)
\(\chi_{16055}(1523,\cdot)\)
\(\chi_{16055}(1857,\cdot)\)
\(\chi_{16055}(1922,\cdot)\)
\(\chi_{16055}(1978,\cdot)\)
\(\chi_{16055}(2008,\cdot)\)
\(\chi_{16055}(2138,\cdot)\)
\(\chi_{16055}(2217,\cdot)\)
\(\chi_{16055}(2333,\cdot)\)
\(\chi_{16055}(2377,\cdot)\)
\(\chi_{16055}(2542,\cdot)\)
\(\chi_{16055}(2693,\cdot)\)
\(\chi_{16055}(2758,\cdot)\)
\(\chi_{16055}(3092,\cdot)\)
\(\chi_{16055}(3157,\cdot)\)
\(\chi_{16055}(3213,\cdot)\)
\(\chi_{16055}(3243,\cdot)\)
...
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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 };
\((3212,14536,14366)\) → \((-i,e\left(\frac{149}{156}\right),e\left(\frac{11}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 16055 }(2333, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{37}{117}\right)\) | \(e\left(\frac{295}{468}\right)\) | \(e\left(\frac{74}{117}\right)\) | \(e\left(\frac{443}{468}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{61}{234}\right)\) | \(e\left(\frac{37}{52}\right)\) | \(e\left(\frac{41}{156}\right)\) | \(e\left(\frac{109}{117}\right)\) |
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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)
