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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5963, base_ring=CyclotomicField(132))
M = H._module
chi = DirichletCharacter(H, M([38,45]))
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gp:[g,chi] = znchar(Mod(1286, 5963))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5963.1286");
| Modulus: | \(5963\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5963\) |
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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: | 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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\(\chi_{5963}(409,\cdot)\)
\(\chi_{5963}(463,\cdot)\)
\(\chi_{5963}(783,\cdot)\)
\(\chi_{5963}(1256,\cdot)\)
\(\chi_{5963}(1286,\cdot)\)
\(\chi_{5963}(1553,\cdot)\)
\(\chi_{5963}(1585,\cdot)\)
\(\chi_{5963}(1686,\cdot)\)
\(\chi_{5963}(1744,\cdot)\)
\(\chi_{5963}(1816,\cdot)\)
\(\chi_{5963}(1859,\cdot)\)
\(\chi_{5963}(2030,\cdot)\)
\(\chi_{5963}(2178,\cdot)\)
\(\chi_{5963}(2335,\cdot)\)
\(\chi_{5963}(2741,\cdot)\)
\(\chi_{5963}(2932,\cdot)\)
\(\chi_{5963}(3005,\cdot)\)
\(\chi_{5963}(3046,\cdot)\)
\(\chi_{5963}(3047,\cdot)\)
\(\chi_{5963}(3151,\cdot)\)
\(\chi_{5963}(3480,\cdot)\)
\(\chi_{5963}(3518,\cdot)\)
\(\chi_{5963}(3569,\cdot)\)
\(\chi_{5963}(3659,\cdot)\)
\(\chi_{5963}(3698,\cdot)\)
\(\chi_{5963}(3717,\cdot)\)
\(\chi_{5963}(3847,\cdot)\)
\(\chi_{5963}(3965,\cdot)\)
\(\chi_{5963}(4252,\cdot)\)
\(\chi_{5963}(4262,\cdot)\)
...
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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 };
\((5697,537)\) → \((e\left(\frac{19}{66}\right),e\left(\frac{15}{44}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 5963 }(1286, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{41}{132}\right)\) | \(e\left(\frac{31}{132}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{41}{66}\right)\) |
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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)
