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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(575, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([33,80]))
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gp:[g,chi] = znchar(Mod(164, 575))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("575.164");
| Modulus: | \(575\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(575\) |
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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: | even |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{575}(4,\cdot)\)
\(\chi_{575}(9,\cdot)\)
\(\chi_{575}(29,\cdot)\)
\(\chi_{575}(39,\cdot)\)
\(\chi_{575}(54,\cdot)\)
\(\chi_{575}(59,\cdot)\)
\(\chi_{575}(64,\cdot)\)
\(\chi_{575}(94,\cdot)\)
\(\chi_{575}(104,\cdot)\)
\(\chi_{575}(119,\cdot)\)
\(\chi_{575}(144,\cdot)\)
\(\chi_{575}(154,\cdot)\)
\(\chi_{575}(164,\cdot)\)
\(\chi_{575}(169,\cdot)\)
\(\chi_{575}(179,\cdot)\)
\(\chi_{575}(209,\cdot)\)
\(\chi_{575}(219,\cdot)\)
\(\chi_{575}(234,\cdot)\)
\(\chi_{575}(239,\cdot)\)
\(\chi_{575}(259,\cdot)\)
\(\chi_{575}(269,\cdot)\)
\(\chi_{575}(279,\cdot)\)
\(\chi_{575}(284,\cdot)\)
\(\chi_{575}(289,\cdot)\)
\(\chi_{575}(294,\cdot)\)
\(\chi_{575}(334,\cdot)\)
\(\chi_{575}(354,\cdot)\)
\(\chi_{575}(384,\cdot)\)
\(\chi_{575}(394,\cdot)\)
\(\chi_{575}(404,\cdot)\)
...
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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()
|
\((277,51)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{8}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 575 }(164, a) \) |
\(1\) | \(1\) | \(e\left(\frac{83}{110}\right)\) | \(e\left(\frac{81}{110}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{97}{110}\right)\) |
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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)
