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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(14455, base_ring=CyclotomicField(1218))
M = H._module
chi = DirichletCharacter(H, M([609,986,1134]))
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gp:[g,chi] = znchar(Mod(284, 14455))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("14455.284");
| Modulus: | \(14455\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(14455\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1218\) |
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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);
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\(\chi_{14455}(4,\cdot)\)
\(\chi_{14455}(9,\cdot)\)
\(\chi_{14455}(74,\cdot)\)
\(\chi_{14455}(144,\cdot)\)
\(\chi_{14455}(184,\cdot)\)
\(\chi_{14455}(284,\cdot)\)
\(\chi_{14455}(289,\cdot)\)
\(\chi_{14455}(359,\cdot)\)
\(\chi_{14455}(389,\cdot)\)
\(\chi_{14455}(429,\cdot)\)
\(\chi_{14455}(464,\cdot)\)
\(\chi_{14455}(494,\cdot)\)
\(\chi_{14455}(499,\cdot)\)
\(\chi_{14455}(529,\cdot)\)
\(\chi_{14455}(534,\cdot)\)
\(\chi_{14455}(599,\cdot)\)
\(\chi_{14455}(639,\cdot)\)
\(\chi_{14455}(669,\cdot)\)
\(\chi_{14455}(674,\cdot)\)
\(\chi_{14455}(744,\cdot)\)
\(\chi_{14455}(774,\cdot)\)
\(\chi_{14455}(779,\cdot)\)
\(\chi_{14455}(879,\cdot)\)
\(\chi_{14455}(914,\cdot)\)
\(\chi_{14455}(989,\cdot)\)
\(\chi_{14455}(1019,\cdot)\)
\(\chi_{14455}(1024,\cdot)\)
\(\chi_{14455}(1054,\cdot)\)
\(\chi_{14455}(1089,\cdot)\)
\(\chi_{14455}(1124,\cdot)\)
...
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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 };
\((2892,591,11271)\) → \((-1,e\left(\frac{17}{21}\right),e\left(\frac{27}{29}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 14455 }(284, a) \) |
\(1\) | \(1\) | \(e\left(\frac{583}{1218}\right)\) | \(e\left(\frac{1049}{1218}\right)\) | \(e\left(\frac{583}{609}\right)\) | \(e\left(\frac{69}{203}\right)\) | \(e\left(\frac{177}{406}\right)\) | \(e\left(\frac{440}{609}\right)\) | \(e\left(\frac{400}{609}\right)\) | \(e\left(\frac{997}{1218}\right)\) | \(e\left(\frac{45}{406}\right)\) | \(e\left(\frac{557}{609}\right)\) |
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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)
