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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(137, base_ring=CyclotomicField(136))
M = H._module
chi = DirichletCharacter(H, M([131]))
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gp:[g,chi] = znchar(Mod(53, 137))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("137.53");
| Modulus: | \(137\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(137\) |
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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: | 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: | odd |
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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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\(\chi_{137}(3,\cdot)\)
\(\chi_{137}(5,\cdot)\)
\(\chi_{137}(6,\cdot)\)
\(\chi_{137}(12,\cdot)\)
\(\chi_{137}(13,\cdot)\)
\(\chi_{137}(20,\cdot)\)
\(\chi_{137}(21,\cdot)\)
\(\chi_{137}(23,\cdot)\)
\(\chi_{137}(24,\cdot)\)
\(\chi_{137}(26,\cdot)\)
\(\chi_{137}(27,\cdot)\)
\(\chi_{137}(29,\cdot)\)
\(\chi_{137}(31,\cdot)\)
\(\chi_{137}(33,\cdot)\)
\(\chi_{137}(35,\cdot)\)
\(\chi_{137}(40,\cdot)\)
\(\chi_{137}(42,\cdot)\)
\(\chi_{137}(43,\cdot)\)
\(\chi_{137}(45,\cdot)\)
\(\chi_{137}(46,\cdot)\)
\(\chi_{137}(47,\cdot)\)
\(\chi_{137}(48,\cdot)\)
\(\chi_{137}(51,\cdot)\)
\(\chi_{137}(52,\cdot)\)
\(\chi_{137}(53,\cdot)\)
\(\chi_{137}(54,\cdot)\)
\(\chi_{137}(55,\cdot)\)
\(\chi_{137}(57,\cdot)\)
\(\chi_{137}(58,\cdot)\)
\(\chi_{137}(62,\cdot)\)
...
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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()
|
\(3\) → \(e\left(\frac{131}{136}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 137 }(53, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{43}{68}\right)\) | \(e\left(\frac{131}{136}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{33}{136}\right)\) | \(e\left(\frac{81}{136}\right)\) | \(e\left(\frac{31}{68}\right)\) | \(e\left(\frac{61}{68}\right)\) | \(e\left(\frac{63}{68}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{35}{68}\right)\) |
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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)
