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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2492, base_ring=CyclotomicField(264))
M = H._module
chi = DirichletCharacter(H, M([132,88,9]))
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gp:[g,chi] = znchar(Mod(1451, 2492))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2492.1451");
| Modulus: | \(2492\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2492\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(264\) |
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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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\(\chi_{2492}(23,\cdot)\)
\(\chi_{2492}(51,\cdot)\)
\(\chi_{2492}(95,\cdot)\)
\(\chi_{2492}(135,\cdot)\)
\(\chi_{2492}(151,\cdot)\)
\(\chi_{2492}(163,\cdot)\)
\(\chi_{2492}(191,\cdot)\)
\(\chi_{2492}(207,\cdot)\)
\(\chi_{2492}(219,\cdot)\)
\(\chi_{2492}(291,\cdot)\)
\(\chi_{2492}(359,\cdot)\)
\(\chi_{2492}(375,\cdot)\)
\(\chi_{2492}(387,\cdot)\)
\(\chi_{2492}(415,\cdot)\)
\(\chi_{2492}(431,\cdot)\)
\(\chi_{2492}(459,\cdot)\)
\(\chi_{2492}(471,\cdot)\)
\(\chi_{2492}(499,\cdot)\)
\(\chi_{2492}(515,\cdot)\)
\(\chi_{2492}(527,\cdot)\)
\(\chi_{2492}(599,\cdot)\)
\(\chi_{2492}(683,\cdot)\)
\(\chi_{2492}(739,\cdot)\)
\(\chi_{2492}(795,\cdot)\)
\(\chi_{2492}(807,\cdot)\)
\(\chi_{2492}(863,\cdot)\)
\(\chi_{2492}(919,\cdot)\)
\(\chi_{2492}(1003,\cdot)\)
\(\chi_{2492}(1075,\cdot)\)
\(\chi_{2492}(1087,\cdot)\)
...
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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 };
\((1247,1781,1961)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{3}{88}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 2492 }(1451, a) \) |
\(1\) | \(1\) | \(e\left(\frac{229}{264}\right)\) | \(e\left(\frac{7}{132}\right)\) | \(e\left(\frac{97}{132}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{69}{88}\right)\) | \(e\left(\frac{81}{88}\right)\) | \(e\left(\frac{71}{132}\right)\) | \(e\left(\frac{95}{264}\right)\) | \(e\left(\frac{29}{264}\right)\) | \(e\left(\frac{7}{66}\right)\) |
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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)
