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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2488, base_ring=CyclotomicField(310))
M = H._module
chi = DirichletCharacter(H, M([155,155,121]))
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gp:[g,chi] = znchar(Mod(1179, 2488))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2488.1179");
| Modulus: | \(2488\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2488\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(310\) |
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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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\(\chi_{2488}(19,\cdot)\)
\(\chi_{2488}(43,\cdot)\)
\(\chi_{2488}(59,\cdot)\)
\(\chi_{2488}(99,\cdot)\)
\(\chi_{2488}(115,\cdot)\)
\(\chi_{2488}(123,\cdot)\)
\(\chi_{2488}(131,\cdot)\)
\(\chi_{2488}(155,\cdot)\)
\(\chi_{2488}(203,\cdot)\)
\(\chi_{2488}(211,\cdot)\)
\(\chi_{2488}(227,\cdot)\)
\(\chi_{2488}(251,\cdot)\)
\(\chi_{2488}(283,\cdot)\)
\(\chi_{2488}(299,\cdot)\)
\(\chi_{2488}(307,\cdot)\)
\(\chi_{2488}(355,\cdot)\)
\(\chi_{2488}(387,\cdot)\)
\(\chi_{2488}(403,\cdot)\)
\(\chi_{2488}(435,\cdot)\)
\(\chi_{2488}(443,\cdot)\)
\(\chi_{2488}(459,\cdot)\)
\(\chi_{2488}(475,\cdot)\)
\(\chi_{2488}(483,\cdot)\)
\(\chi_{2488}(515,\cdot)\)
\(\chi_{2488}(547,\cdot)\)
\(\chi_{2488}(555,\cdot)\)
\(\chi_{2488}(587,\cdot)\)
\(\chi_{2488}(595,\cdot)\)
\(\chi_{2488}(619,\cdot)\)
\(\chi_{2488}(651,\cdot)\)
...
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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 };
\((623,1245,17)\) → \((-1,-1,e\left(\frac{121}{310}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 2488 }(1179, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{155}\right)\) | \(e\left(\frac{1}{310}\right)\) | \(e\left(\frac{9}{62}\right)\) | \(e\left(\frac{4}{155}\right)\) | \(e\left(\frac{43}{62}\right)\) | \(e\left(\frac{45}{62}\right)\) | \(e\left(\frac{1}{62}\right)\) | \(e\left(\frac{121}{310}\right)\) | \(e\left(\frac{247}{310}\right)\) | \(e\left(\frac{49}{310}\right)\) |
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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)
