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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(578, base_ring=CyclotomicField(136))
M = H._module
chi = DirichletCharacter(H, M([19]))
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gp:[g,chi] = znchar(Mod(49, 578))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("578.49");
| Modulus: | \(578\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(289\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(136\) |
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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: | no, induced from \(\chi_{289}(49,\cdot)\) |
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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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\(\chi_{578}(9,\cdot)\)
\(\chi_{578}(15,\cdot)\)
\(\chi_{578}(19,\cdot)\)
\(\chi_{578}(25,\cdot)\)
\(\chi_{578}(43,\cdot)\)
\(\chi_{578}(49,\cdot)\)
\(\chi_{578}(53,\cdot)\)
\(\chi_{578}(59,\cdot)\)
\(\chi_{578}(77,\cdot)\)
\(\chi_{578}(83,\cdot)\)
\(\chi_{578}(87,\cdot)\)
\(\chi_{578}(93,\cdot)\)
\(\chi_{578}(111,\cdot)\)
\(\chi_{578}(117,\cdot)\)
\(\chi_{578}(121,\cdot)\)
\(\chi_{578}(127,\cdot)\)
\(\chi_{578}(145,\cdot)\)
\(\chi_{578}(151,\cdot)\)
\(\chi_{578}(161,\cdot)\)
\(\chi_{578}(185,\cdot)\)
\(\chi_{578}(189,\cdot)\)
\(\chi_{578}(195,\cdot)\)
\(\chi_{578}(213,\cdot)\)
\(\chi_{578}(219,\cdot)\)
\(\chi_{578}(223,\cdot)\)
\(\chi_{578}(229,\cdot)\)
\(\chi_{578}(247,\cdot)\)
\(\chi_{578}(253,\cdot)\)
\(\chi_{578}(257,\cdot)\)
\(\chi_{578}(263,\cdot)\)
...
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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 };
| Field of values: |
$\Q(\zeta_{136})$ |
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sage:CyclotomicField(chi.multiplicative_order())
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gp:nfinit(polcyclo(charorder(g,chi)))
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magma:CyclotomicField(Order(chi));
|
| Fixed field: |
Number field defined by a degree 136 polynomial (not computed) |
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sage:chi.fixed_field()
|
\(3\) → \(e\left(\frac{19}{136}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 578 }(49, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{136}\right)\) | \(e\left(\frac{135}{136}\right)\) | \(e\left(\frac{89}{136}\right)\) | \(e\left(\frac{19}{68}\right)\) | \(e\left(\frac{29}{136}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{9}{68}\right)\) | \(e\left(\frac{65}{68}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{21}{136}\right)\) |
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sage:chi(x) # x integer
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gp:chareval(g,chi,x) \\ x integer, value in Q/Z
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magma:chi(x)
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sage:chi.gauss_sum(a)
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gp:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
