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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(14365, base_ring=CyclotomicField(104))
M = H._module
chi = DirichletCharacter(H, M([78,94,39]))
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gp:[g,chi] = znchar(Mod(6713, 14365))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("14365.6713");
| Modulus: | \(14365\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(14365\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(104\) |
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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);
|
\(\chi_{14365}(83,\cdot)\)
\(\chi_{14365}(502,\cdot)\)
\(\chi_{14365}(733,\cdot)\)
\(\chi_{14365}(892,\cdot)\)
\(\chi_{14365}(1188,\cdot)\)
\(\chi_{14365}(1607,\cdot)\)
\(\chi_{14365}(1838,\cdot)\)
\(\chi_{14365}(1997,\cdot)\)
\(\chi_{14365}(2293,\cdot)\)
\(\chi_{14365}(2712,\cdot)\)
\(\chi_{14365}(3102,\cdot)\)
\(\chi_{14365}(3398,\cdot)\)
\(\chi_{14365}(4048,\cdot)\)
\(\chi_{14365}(4207,\cdot)\)
\(\chi_{14365}(4503,\cdot)\)
\(\chi_{14365}(4922,\cdot)\)
\(\chi_{14365}(5153,\cdot)\)
\(\chi_{14365}(5312,\cdot)\)
\(\chi_{14365}(5608,\cdot)\)
\(\chi_{14365}(6027,\cdot)\)
\(\chi_{14365}(6258,\cdot)\)
\(\chi_{14365}(6417,\cdot)\)
\(\chi_{14365}(6713,\cdot)\)
\(\chi_{14365}(7132,\cdot)\)
\(\chi_{14365}(7363,\cdot)\)
\(\chi_{14365}(7522,\cdot)\)
\(\chi_{14365}(7818,\cdot)\)
\(\chi_{14365}(8237,\cdot)\)
\(\chi_{14365}(8468,\cdot)\)
\(\chi_{14365}(8627,\cdot)\)
...
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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 };
\((5747,171,2536)\) → \((-i,e\left(\frac{47}{52}\right),e\left(\frac{3}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 14365 }(6713, a) \) |
\(1\) | \(1\) | \(e\left(\frac{47}{52}\right)\) | \(e\left(\frac{73}{104}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{63}{104}\right)\) | \(e\left(\frac{61}{104}\right)\) | \(e\left(\frac{37}{52}\right)\) | \(e\left(\frac{21}{52}\right)\) | \(e\left(\frac{75}{104}\right)\) | \(e\left(\frac{53}{104}\right)\) | \(e\left(\frac{51}{104}\right)\) |
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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)
