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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(11431, base_ring=CyclotomicField(462))
M = H._module
chi = DirichletCharacter(H, M([154,420,297]))
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gp:[g,chi] = znchar(Mod(2795, 11431))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("11431.2795");
| Modulus: | \(11431\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(11431\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(462\) |
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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);
|
\(\chi_{11431}(39,\cdot)\)
\(\chi_{11431}(165,\cdot)\)
\(\chi_{11431}(193,\cdot)\)
\(\chi_{11431}(394,\cdot)\)
\(\chi_{11431}(662,\cdot)\)
\(\chi_{11431}(744,\cdot)\)
\(\chi_{11431}(807,\cdot)\)
\(\chi_{11431}(886,\cdot)\)
\(\chi_{11431}(949,\cdot)\)
\(\chi_{11431}(1159,\cdot)\)
\(\chi_{11431}(1248,\cdot)\)
\(\chi_{11431}(1304,\cdot)\)
\(\chi_{11431}(1383,\cdot)\)
\(\chi_{11431}(1388,\cdot)\)
\(\chi_{11431}(1530,\cdot)\)
\(\chi_{11431}(1738,\cdot)\)
\(\chi_{11431}(2027,\cdot)\)
\(\chi_{11431}(2235,\cdot)\)
\(\chi_{11431}(2377,\cdot)\)
\(\chi_{11431}(2382,\cdot)\)
\(\chi_{11431}(2440,\cdot)\)
\(\chi_{11431}(2536,\cdot)\)
\(\chi_{11431}(2732,\cdot)\)
\(\chi_{11431}(2739,\cdot)\)
\(\chi_{11431}(2795,\cdot)\)
\(\chi_{11431}(2879,\cdot)\)
\(\chi_{11431}(2881,\cdot)\)
\(\chi_{11431}(2937,\cdot)\)
\(\chi_{11431}(3021,\cdot)\)
\(\chi_{11431}(3229,\cdot)\)
...
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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 };
\((1634,9941,3060)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{10}{11}\right),e\left(\frac{9}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 11431 }(2795, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{79}{231}\right)\) | \(e\left(\frac{137}{231}\right)\) | \(e\left(\frac{158}{231}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{72}{77}\right)\) | \(e\left(\frac{2}{77}\right)\) | \(e\left(\frac{43}{231}\right)\) | \(e\left(\frac{212}{231}\right)\) | \(e\left(\frac{205}{462}\right)\) | \(e\left(\frac{64}{231}\right)\) |
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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)
