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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2452, base_ring=CyclotomicField(612))
M = H._module
chi = DirichletCharacter(H, M([306,293]))
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gp:[g,chi] = znchar(Mod(1215, 2452))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2452.1215");
| Modulus: | \(2452\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2452\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(612\) |
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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);
|
\(\chi_{2452}(11,\cdot)\)
\(\chi_{2452}(15,\cdot)\)
\(\chi_{2452}(91,\cdot)\)
\(\chi_{2452}(95,\cdot)\)
\(\chi_{2452}(99,\cdot)\)
\(\chi_{2452}(127,\cdot)\)
\(\chi_{2452}(135,\cdot)\)
\(\chi_{2452}(159,\cdot)\)
\(\chi_{2452}(163,\cdot)\)
\(\chi_{2452}(211,\cdot)\)
\(\chi_{2452}(215,\cdot)\)
\(\chi_{2452}(219,\cdot)\)
\(\chi_{2452}(223,\cdot)\)
\(\chi_{2452}(235,\cdot)\)
\(\chi_{2452}(247,\cdot)\)
\(\chi_{2452}(251,\cdot)\)
\(\chi_{2452}(271,\cdot)\)
\(\chi_{2452}(283,\cdot)\)
\(\chi_{2452}(307,\cdot)\)
\(\chi_{2452}(311,\cdot)\)
\(\chi_{2452}(327,\cdot)\)
\(\chi_{2452}(331,\cdot)\)
\(\chi_{2452}(335,\cdot)\)
\(\chi_{2452}(339,\cdot)\)
\(\chi_{2452}(347,\cdot)\)
\(\chi_{2452}(351,\cdot)\)
\(\chi_{2452}(355,\cdot)\)
\(\chi_{2452}(371,\cdot)\)
\(\chi_{2452}(391,\cdot)\)
\(\chi_{2452}(439,\cdot)\)
...
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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 };
\((1227,1841)\) → \((-1,e\left(\frac{293}{612}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 2452 }(1215, a) \) |
\(1\) | \(1\) | \(e\left(\frac{4}{51}\right)\) | \(e\left(\frac{235}{612}\right)\) | \(e\left(\frac{265}{306}\right)\) | \(e\left(\frac{8}{51}\right)\) | \(e\left(\frac{169}{612}\right)\) | \(e\left(\frac{173}{612}\right)\) | \(e\left(\frac{283}{612}\right)\) | \(e\left(\frac{59}{153}\right)\) | \(e\left(\frac{101}{102}\right)\) | \(e\left(\frac{17}{18}\right)\) |
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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)
