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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5741, base_ring=CyclotomicField(2870))
M = H._module
chi = DirichletCharacter(H, M([2446]))
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gp:[g,chi] = znchar(Mod(49, 5741))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5741.49");
| Modulus: | \(5741\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5741\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1435\) |
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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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\(\chi_{5741}(5,\cdot)\)
\(\chi_{5741}(16,\cdot)\)
\(\chi_{5741}(22,\cdot)\)
\(\chi_{5741}(25,\cdot)\)
\(\chi_{5741}(26,\cdot)\)
\(\chi_{5741}(28,\cdot)\)
\(\chi_{5741}(33,\cdot)\)
\(\chi_{5741}(36,\cdot)\)
\(\chi_{5741}(49,\cdot)\)
\(\chi_{5741}(63,\cdot)\)
\(\chi_{5741}(81,\cdot)\)
\(\chi_{5741}(86,\cdot)\)
\(\chi_{5741}(97,\cdot)\)
\(\chi_{5741}(110,\cdot)\)
\(\chi_{5741}(120,\cdot)\)
\(\chi_{5741}(122,\cdot)\)
\(\chi_{5741}(125,\cdot)\)
\(\chi_{5741}(127,\cdot)\)
\(\chi_{5741}(129,\cdot)\)
\(\chi_{5741}(130,\cdot)\)
\(\chi_{5741}(134,\cdot)\)
\(\chi_{5741}(139,\cdot)\)
\(\chi_{5741}(140,\cdot)\)
\(\chi_{5741}(152,\cdot)\)
\(\chi_{5741}(157,\cdot)\)
\(\chi_{5741}(158,\cdot)\)
\(\chi_{5741}(163,\cdot)\)
\(\chi_{5741}(167,\cdot)\)
\(\chi_{5741}(174,\cdot)\)
\(\chi_{5741}(183,\cdot)\)
...
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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 };
\(2\) → \(e\left(\frac{1223}{1435}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 5741 }(49, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1223}{1435}\right)\) | \(e\left(\frac{244}{1435}\right)\) | \(e\left(\frac{1011}{1435}\right)\) | \(e\left(\frac{888}{1435}\right)\) | \(e\left(\frac{32}{1435}\right)\) | \(e\left(\frac{918}{1435}\right)\) | \(e\left(\frac{799}{1435}\right)\) | \(e\left(\frac{488}{1435}\right)\) | \(e\left(\frac{676}{1435}\right)\) | \(e\left(\frac{200}{287}\right)\) |
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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)
