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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(371, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([26,51]))
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gp:[g,chi] = znchar(Mod(3, 371))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("371.3");
| Modulus: | \(371\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(371\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(156\) |
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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);
|
\(\chi_{371}(3,\cdot)\)
\(\chi_{371}(5,\cdot)\)
\(\chi_{371}(12,\cdot)\)
\(\chi_{371}(19,\cdot)\)
\(\chi_{371}(26,\cdot)\)
\(\chi_{371}(31,\cdot)\)
\(\chi_{371}(33,\cdot)\)
\(\chi_{371}(45,\cdot)\)
\(\chi_{371}(61,\cdot)\)
\(\chi_{371}(73,\cdot)\)
\(\chi_{371}(75,\cdot)\)
\(\chi_{371}(80,\cdot)\)
\(\chi_{371}(87,\cdot)\)
\(\chi_{371}(94,\cdot)\)
\(\chi_{371}(101,\cdot)\)
\(\chi_{371}(103,\cdot)\)
\(\chi_{371}(108,\cdot)\)
\(\chi_{371}(124,\cdot)\)
\(\chi_{371}(138,\cdot)\)
\(\chi_{371}(145,\cdot)\)
\(\chi_{371}(157,\cdot)\)
\(\chi_{371}(164,\cdot)\)
\(\chi_{371}(171,\cdot)\)
\(\chi_{371}(173,\cdot)\)
\(\chi_{371}(178,\cdot)\)
\(\chi_{371}(180,\cdot)\)
\(\chi_{371}(185,\cdot)\)
\(\chi_{371}(192,\cdot)\)
\(\chi_{371}(194,\cdot)\)
\(\chi_{371}(215,\cdot)\)
...
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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: |
$\Q(\zeta_{156})$ |
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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 156 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((213,267)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{17}{52}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 371 }(3, a) \) |
\(1\) | \(1\) | \(e\left(\frac{103}{156}\right)\) | \(e\left(\frac{113}{156}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{31}{156}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{51}{52}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{7}{156}\right)\) |
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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)
