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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8049, base_ring=CyclotomicField(2682))
M = H._module
chi = DirichletCharacter(H, M([1341,2546]))
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gp:[g,chi] = znchar(Mod(95, 8049))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8049.95");
| Modulus: | \(8049\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(8049\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(2682\) |
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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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\(\chi_{8049}(11,\cdot)\)
\(\chi_{8049}(14,\cdot)\)
\(\chi_{8049}(23,\cdot)\)
\(\chi_{8049}(26,\cdot)\)
\(\chi_{8049}(35,\cdot)\)
\(\chi_{8049}(38,\cdot)\)
\(\chi_{8049}(41,\cdot)\)
\(\chi_{8049}(65,\cdot)\)
\(\chi_{8049}(71,\cdot)\)
\(\chi_{8049}(74,\cdot)\)
\(\chi_{8049}(95,\cdot)\)
\(\chi_{8049}(107,\cdot)\)
\(\chi_{8049}(110,\cdot)\)
\(\chi_{8049}(140,\cdot)\)
\(\chi_{8049}(158,\cdot)\)
\(\chi_{8049}(173,\cdot)\)
\(\chi_{8049}(176,\cdot)\)
\(\chi_{8049}(215,\cdot)\)
\(\chi_{8049}(218,\cdot)\)
\(\chi_{8049}(224,\cdot)\)
\(\chi_{8049}(230,\cdot)\)
\(\chi_{8049}(233,\cdot)\)
\(\chi_{8049}(254,\cdot)\)
\(\chi_{8049}(257,\cdot)\)
\(\chi_{8049}(260,\cdot)\)
\(\chi_{8049}(275,\cdot)\)
\(\chi_{8049}(284,\cdot)\)
\(\chi_{8049}(296,\cdot)\)
\(\chi_{8049}(329,\cdot)\)
\(\chi_{8049}(341,\cdot)\)
...
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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 };
\((2684,5368)\) → \((-1,e\left(\frac{1273}{1341}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 8049 }(95, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1205}{2682}\right)\) | \(e\left(\frac{1205}{1341}\right)\) | \(e\left(\frac{1105}{2682}\right)\) | \(e\left(\frac{320}{447}\right)\) | \(e\left(\frac{311}{894}\right)\) | \(e\left(\frac{385}{447}\right)\) | \(e\left(\frac{1625}{2682}\right)\) | \(e\left(\frac{35}{149}\right)\) | \(e\left(\frac{443}{2682}\right)\) | \(e\left(\frac{1069}{1341}\right)\) |
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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)
