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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(269, base_ring=CyclotomicField(268))
M = H._module
chi = DirichletCharacter(H, M([129]))
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pari:[g,chi] = znchar(Mod(42,269))
| Modulus: | \(269\) | |
| Conductor: | \(269\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(268\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
|
| Real: | no |
| Primitive: | yes |
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sage:chi.is_primitive()
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pari:#znconreyconductor(g,chi)==1
|
| Minimal: | yes |
| Parity: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{269}(2,\cdot)\)
\(\chi_{269}(3,\cdot)\)
\(\chi_{269}(7,\cdot)\)
\(\chi_{269}(8,\cdot)\)
\(\chi_{269}(10,\cdot)\)
\(\chi_{269}(12,\cdot)\)
\(\chi_{269}(15,\cdot)\)
\(\chi_{269}(17,\cdot)\)
\(\chi_{269}(18,\cdot)\)
\(\chi_{269}(19,\cdot)\)
\(\chi_{269}(22,\cdot)\)
\(\chi_{269}(26,\cdot)\)
\(\chi_{269}(27,\cdot)\)
\(\chi_{269}(28,\cdot)\)
\(\chi_{269}(29,\cdot)\)
\(\chi_{269}(31,\cdot)\)
\(\chi_{269}(32,\cdot)\)
\(\chi_{269}(33,\cdot)\)
\(\chi_{269}(35,\cdot)\)
\(\chi_{269}(39,\cdot)\)
\(\chi_{269}(40,\cdot)\)
\(\chi_{269}(42,\cdot)\)
\(\chi_{269}(46,\cdot)\)
\(\chi_{269}(48,\cdot)\)
\(\chi_{269}(50,\cdot)\)
\(\chi_{269}(59,\cdot)\)
\(\chi_{269}(60,\cdot)\)
\(\chi_{269}(63,\cdot)\)
\(\chi_{269}(68,\cdot)\)
\(\chi_{269}(69,\cdot)\)
...
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sage:chi.galois_orbit()
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pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\(2\) → \(e\left(\frac{129}{268}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 269 }(42, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{129}{268}\right)\) | \(e\left(\frac{125}{268}\right)\) | \(e\left(\frac{129}{134}\right)\) | \(e\left(\frac{8}{67}\right)\) | \(e\left(\frac{127}{134}\right)\) | \(e\left(\frac{39}{268}\right)\) | \(e\left(\frac{119}{268}\right)\) | \(e\left(\frac{125}{134}\right)\) | \(e\left(\frac{161}{268}\right)\) | \(e\left(\frac{95}{134}\right)\) |
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sage:chi.jacobi_sum(n)
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sage:chi.gauss_sum(a)
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pari:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
