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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2259, base_ring=CyclotomicField(150))
M = H._module
chi = DirichletCharacter(H, M([100,81]))
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gp:[g,chi] = znchar(Mod(979, 2259))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2259.979");
| Modulus: | \(2259\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2259\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(150\) |
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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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\(\chi_{2259}(40,\cdot)\)
\(\chi_{2259}(151,\cdot)\)
\(\chi_{2259}(157,\cdot)\)
\(\chi_{2259}(160,\cdot)\)
\(\chi_{2259}(187,\cdot)\)
\(\chi_{2259}(247,\cdot)\)
\(\chi_{2259}(259,\cdot)\)
\(\chi_{2259}(301,\cdot)\)
\(\chi_{2259}(439,\cdot)\)
\(\chi_{2259}(628,\cdot)\)
\(\chi_{2259}(673,\cdot)\)
\(\chi_{2259}(763,\cdot)\)
\(\chi_{2259}(904,\cdot)\)
\(\chi_{2259}(913,\cdot)\)
\(\chi_{2259}(940,\cdot)\)
\(\chi_{2259}(979,\cdot)\)
\(\chi_{2259}(988,\cdot)\)
\(\chi_{2259}(1006,\cdot)\)
\(\chi_{2259}(1012,\cdot)\)
\(\chi_{2259}(1051,\cdot)\)
\(\chi_{2259}(1132,\cdot)\)
\(\chi_{2259}(1186,\cdot)\)
\(\chi_{2259}(1192,\cdot)\)
\(\chi_{2259}(1204,\cdot)\)
\(\chi_{2259}(1381,\cdot)\)
\(\chi_{2259}(1426,\cdot)\)
\(\chi_{2259}(1501,\cdot)\)
\(\chi_{2259}(1516,\cdot)\)
\(\chi_{2259}(1546,\cdot)\)
\(\chi_{2259}(1663,\cdot)\)
...
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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 };
\((2009,1261)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{27}{50}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 2259 }(979, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{44}{75}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{91}{150}\right)\) | \(e\left(\frac{46}{75}\right)\) | \(e\left(\frac{23}{150}\right)\) | \(e\left(\frac{4}{15}\right)\) |
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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)
