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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(869, base_ring=CyclotomicField(130))
M = H._module
chi = DirichletCharacter(H, M([13,120]))
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gp:[g,chi] = znchar(Mod(101, 869))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("869.101");
| Modulus: | \(869\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(869\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(130\) |
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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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\(\chi_{869}(8,\cdot)\)
\(\chi_{869}(18,\cdot)\)
\(\chi_{869}(46,\cdot)\)
\(\chi_{869}(52,\cdot)\)
\(\chi_{869}(62,\cdot)\)
\(\chi_{869}(101,\cdot)\)
\(\chi_{869}(117,\cdot)\)
\(\chi_{869}(204,\cdot)\)
\(\chi_{869}(222,\cdot)\)
\(\chi_{869}(255,\cdot)\)
\(\chi_{869}(259,\cdot)\)
\(\chi_{869}(283,\cdot)\)
\(\chi_{869}(299,\cdot)\)
\(\chi_{869}(304,\cdot)\)
\(\chi_{869}(326,\cdot)\)
\(\chi_{869}(337,\cdot)\)
\(\chi_{869}(338,\cdot)\)
\(\chi_{869}(354,\cdot)\)
\(\chi_{869}(380,\cdot)\)
\(\chi_{869}(381,\cdot)\)
\(\chi_{869}(403,\cdot)\)
\(\chi_{869}(413,\cdot)\)
\(\chi_{869}(447,\cdot)\)
\(\chi_{869}(457,\cdot)\)
\(\chi_{869}(459,\cdot)\)
\(\chi_{869}(492,\cdot)\)
\(\chi_{869}(512,\cdot)\)
\(\chi_{869}(536,\cdot)\)
\(\chi_{869}(541,\cdot)\)
\(\chi_{869}(563,\cdot)\)
...
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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 };
\((475,793)\) → \((e\left(\frac{1}{10}\right),e\left(\frac{12}{13}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 869 }(101, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{103}{130}\right)\) | \(e\left(\frac{47}{65}\right)\) | \(e\left(\frac{38}{65}\right)\) | \(e\left(\frac{41}{65}\right)\) | \(e\left(\frac{67}{130}\right)\) | \(e\left(\frac{81}{130}\right)\) | \(e\left(\frac{49}{130}\right)\) | \(e\left(\frac{29}{65}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{4}{13}\right)\) |
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sage:chi(x)
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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)
