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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4207, base_ring=CyclotomicField(150))
M = H._module
chi = DirichletCharacter(H, M([50,99]))
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gp:[g,chi] = znchar(Mod(2396, 4207))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4207.2396");
| Modulus: | \(4207\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4207\) |
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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: | 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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\(\chi_{4207}(338,\cdot)\)
\(\chi_{4207}(345,\cdot)\)
\(\chi_{4207}(473,\cdot)\)
\(\chi_{4207}(585,\cdot)\)
\(\chi_{4207}(597,\cdot)\)
\(\chi_{4207}(599,\cdot)\)
\(\chi_{4207}(676,\cdot)\)
\(\chi_{4207}(690,\cdot)\)
\(\chi_{4207}(751,\cdot)\)
\(\chi_{4207}(823,\cdot)\)
\(\chi_{4207}(1045,\cdot)\)
\(\chi_{4207}(1094,\cdot)\)
\(\chi_{4207}(1138,\cdot)\)
\(\chi_{4207}(1194,\cdot)\)
\(\chi_{4207}(1313,\cdot)\)
\(\chi_{4207}(1502,\cdot)\)
\(\chi_{4207}(1558,\cdot)\)
\(\chi_{4207}(1675,\cdot)\)
\(\chi_{4207}(1787,\cdot)\)
\(\chi_{4207}(1801,\cdot)\)
\(\chi_{4207}(1878,\cdot)\)
\(\chi_{4207}(1892,\cdot)\)
\(\chi_{4207}(2025,\cdot)\)
\(\chi_{4207}(2188,\cdot)\)
\(\chi_{4207}(2340,\cdot)\)
\(\chi_{4207}(2377,\cdot)\)
\(\chi_{4207}(2396,\cdot)\)
\(\chi_{4207}(2515,\cdot)\)
\(\chi_{4207}(2704,\cdot)\)
\(\chi_{4207}(2760,\cdot)\)
...
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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 };
\((3006,3613)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{33}{50}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 4207 }(2396, a) \) |
\(1\) | \(1\) | \(e\left(\frac{59}{75}\right)\) | \(e\left(\frac{73}{75}\right)\) | \(e\left(\frac{43}{75}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{71}{75}\right)\) | \(e\left(\frac{34}{75}\right)\) | \(e\left(\frac{143}{150}\right)\) | \(e\left(\frac{41}{75}\right)\) |
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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)
