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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(968, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([55,55,32]))
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gp:[g,chi] = znchar(Mod(59, 968))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("968.59");
| Modulus: | \(968\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(968\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(110\) |
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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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\(\chi_{968}(59,\cdot)\)
\(\chi_{968}(75,\cdot)\)
\(\chi_{968}(91,\cdot)\)
\(\chi_{968}(115,\cdot)\)
\(\chi_{968}(147,\cdot)\)
\(\chi_{968}(163,\cdot)\)
\(\chi_{968}(179,\cdot)\)
\(\chi_{968}(203,\cdot)\)
\(\chi_{968}(235,\cdot)\)
\(\chi_{968}(267,\cdot)\)
\(\chi_{968}(291,\cdot)\)
\(\chi_{968}(339,\cdot)\)
\(\chi_{968}(355,\cdot)\)
\(\chi_{968}(379,\cdot)\)
\(\chi_{968}(411,\cdot)\)
\(\chi_{968}(427,\cdot)\)
\(\chi_{968}(443,\cdot)\)
\(\chi_{968}(467,\cdot)\)
\(\chi_{968}(499,\cdot)\)
\(\chi_{968}(515,\cdot)\)
\(\chi_{968}(531,\cdot)\)
\(\chi_{968}(555,\cdot)\)
\(\chi_{968}(587,\cdot)\)
\(\chi_{968}(603,\cdot)\)
\(\chi_{968}(619,\cdot)\)
\(\chi_{968}(643,\cdot)\)
\(\chi_{968}(675,\cdot)\)
\(\chi_{968}(691,\cdot)\)
\(\chi_{968}(707,\cdot)\)
\(\chi_{968}(731,\cdot)\)
...
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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: |
$\Q(\zeta_{55})$ |
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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 110 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((727,485,849)\) → \((-1,-1,e\left(\frac{16}{55}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 968 }(59, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{59}{110}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{19}{22}\right)\) |
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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)
