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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5723, base_ring=CyclotomicField(1392))
M = H._module
chi = DirichletCharacter(H, M([1200,1015]))
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gp:[g,chi] = znchar(Mod(3, 5723))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5723.3");
| Modulus: | \(5723\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5723\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1392\) |
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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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\(\chi_{5723}(3,\cdot)\)
\(\chi_{5723}(25,\cdot)\)
\(\chi_{5723}(48,\cdot)\)
\(\chi_{5723}(49,\cdot)\)
\(\chi_{5723}(53,\cdot)\)
\(\chi_{5723}(66,\cdot)\)
\(\chi_{5723}(86,\cdot)\)
\(\chi_{5723}(94,\cdot)\)
\(\chi_{5723}(95,\cdot)\)
\(\chi_{5723}(100,\cdot)\)
\(\chi_{5723}(108,\cdot)\)
\(\chi_{5723}(122,\cdot)\)
\(\chi_{5723}(145,\cdot)\)
\(\chi_{5723}(146,\cdot)\)
\(\chi_{5723}(163,\cdot)\)
\(\chi_{5723}(169,\cdot)\)
\(\chi_{5723}(192,\cdot)\)
\(\chi_{5723}(196,\cdot)\)
\(\chi_{5723}(197,\cdot)\)
\(\chi_{5723}(205,\cdot)\)
\(\chi_{5723}(225,\cdot)\)
\(\chi_{5723}(226,\cdot)\)
\(\chi_{5723}(243,\cdot)\)
\(\chi_{5723}(289,\cdot)\)
\(\chi_{5723}(293,\cdot)\)
\(\chi_{5723}(302,\cdot)\)
\(\chi_{5723}(316,\cdot)\)
\(\chi_{5723}(322,\cdot)\)
\(\chi_{5723}(323,\cdot)\)
\(\chi_{5723}(340,\cdot)\)
...
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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 };
\((1359,296)\) → \((e\left(\frac{25}{29}\right),e\left(\frac{35}{48}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 5723 }(3, a) \) |
\(1\) | \(1\) | \(e\left(\frac{455}{696}\right)\) | \(e\left(\frac{101}{696}\right)\) | \(e\left(\frac{107}{348}\right)\) | \(e\left(\frac{1255}{1392}\right)\) | \(e\left(\frac{139}{174}\right)\) | \(e\left(\frac{169}{1392}\right)\) | \(e\left(\frac{223}{232}\right)\) | \(e\left(\frac{101}{348}\right)\) | \(e\left(\frac{773}{1392}\right)\) | \(e\left(\frac{181}{696}\right)\) |
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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)
