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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(66755, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([78,103,44]))
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gp:[g,chi] = znchar(Mod(16404, 66755))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("66755.16404");
| Modulus: | \(66755\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(66755\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(156\) |
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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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\(\chi_{66755}(3369,\cdot)\)
\(\chi_{66755}(3599,\cdot)\)
\(\chi_{66755}(9319,\cdot)\)
\(\chi_{66755}(10004,\cdot)\)
\(\chi_{66755}(11104,\cdot)\)
\(\chi_{66755}(12759,\cdot)\)
\(\chi_{66755}(13314,\cdot)\)
\(\chi_{66755}(13479,\cdot)\)
\(\chi_{66755}(15399,\cdot)\)
\(\chi_{66755}(15489,\cdot)\)
\(\chi_{66755}(15594,\cdot)\)
\(\chi_{66755}(16404,\cdot)\)
\(\chi_{66755}(17314,\cdot)\)
\(\chi_{66755}(17899,\cdot)\)
\(\chi_{66755}(18584,\cdot)\)
\(\chi_{66755}(18649,\cdot)\)
\(\chi_{66755}(19364,\cdot)\)
\(\chi_{66755}(21829,\cdot)\)
\(\chi_{66755}(24264,\cdot)\)
\(\chi_{66755}(24334,\cdot)\)
\(\chi_{66755}(25694,\cdot)\)
\(\chi_{66755}(26639,\cdot)\)
\(\chi_{66755}(26704,\cdot)\)
\(\chi_{66755}(26734,\cdot)\)
\(\chi_{66755}(27839,\cdot)\)
\(\chi_{66755}(30024,\cdot)\)
\(\chi_{66755}(31999,\cdot)\)
\(\chi_{66755}(34539,\cdot)\)
\(\chi_{66755}(35254,\cdot)\)
\(\chi_{66755}(35674,\cdot)\)
...
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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 };
\((13352,33971,38871)\) → \((-1,e\left(\frac{103}{156}\right),e\left(\frac{11}{39}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 66755 }(16404, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{15}{52}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{49}{52}\right)\) | \(e\left(\frac{5}{52}\right)\) | \(e\left(\frac{45}{52}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{29}{156}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) |
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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)
