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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(43681, base_ring=CyclotomicField(6270))
M = H._module
chi = DirichletCharacter(H, M([6042,1100]))
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gp:[g,chi] = znchar(Mod(900, 43681))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("43681.900");
| Modulus: | \(43681\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(43681\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(3135\) |
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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);
|
\(\chi_{43681}(26,\cdot)\)
\(\chi_{43681}(49,\cdot)\)
\(\chi_{43681}(64,\cdot)\)
\(\chi_{43681}(102,\cdot)\)
\(\chi_{43681}(125,\cdot)\)
\(\chi_{43681}(159,\cdot)\)
\(\chi_{43681}(163,\cdot)\)
\(\chi_{43681}(201,\cdot)\)
\(\chi_{43681}(235,\cdot)\)
\(\chi_{43681}(258,\cdot)\)
\(\chi_{43681}(273,\cdot)\)
\(\chi_{43681}(311,\cdot)\)
\(\chi_{43681}(334,\cdot)\)
\(\chi_{43681}(368,\cdot)\)
\(\chi_{43681}(410,\cdot)\)
\(\chi_{43681}(467,\cdot)\)
\(\chi_{43681}(482,\cdot)\)
\(\chi_{43681}(520,\cdot)\)
\(\chi_{43681}(543,\cdot)\)
\(\chi_{43681}(577,\cdot)\)
\(\chi_{43681}(581,\cdot)\)
\(\chi_{43681}(619,\cdot)\)
\(\chi_{43681}(676,\cdot)\)
\(\chi_{43681}(691,\cdot)\)
\(\chi_{43681}(752,\cdot)\)
\(\chi_{43681}(786,\cdot)\)
\(\chi_{43681}(828,\cdot)\)
\(\chi_{43681}(862,\cdot)\)
\(\chi_{43681}(885,\cdot)\)
\(\chi_{43681}(900,\cdot)\)
...
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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 };
\((21661,22023)\) → \((e\left(\frac{53}{55}\right),e\left(\frac{10}{57}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 43681 }(900, a) \) |
\(1\) | \(1\) | \(e\left(\frac{436}{3135}\right)\) | \(e\left(\frac{53}{285}\right)\) | \(e\left(\frac{872}{3135}\right)\) | \(e\left(\frac{34}{3135}\right)\) | \(e\left(\frac{1019}{3135}\right)\) | \(e\left(\frac{64}{1045}\right)\) | \(e\left(\frac{436}{1045}\right)\) | \(e\left(\frac{106}{285}\right)\) | \(e\left(\frac{94}{627}\right)\) | \(e\left(\frac{97}{209}\right)\) |
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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)
