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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(289, base_ring=CyclotomicField(272))
M = H._module
chi = DirichletCharacter(H, M([135]))
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pari:[g,chi] = znchar(Mod(96,289))
| Modulus: | \(289\) | |
| Conductor: | \(289\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(272\) |
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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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\(\chi_{289}(3,\cdot)\)
\(\chi_{289}(5,\cdot)\)
\(\chi_{289}(6,\cdot)\)
\(\chi_{289}(7,\cdot)\)
\(\chi_{289}(10,\cdot)\)
\(\chi_{289}(11,\cdot)\)
\(\chi_{289}(12,\cdot)\)
\(\chi_{289}(14,\cdot)\)
\(\chi_{289}(20,\cdot)\)
\(\chi_{289}(22,\cdot)\)
\(\chi_{289}(23,\cdot)\)
\(\chi_{289}(24,\cdot)\)
\(\chi_{289}(27,\cdot)\)
\(\chi_{289}(28,\cdot)\)
\(\chi_{289}(29,\cdot)\)
\(\chi_{289}(31,\cdot)\)
\(\chi_{289}(37,\cdot)\)
\(\chi_{289}(39,\cdot)\)
\(\chi_{289}(41,\cdot)\)
\(\chi_{289}(44,\cdot)\)
\(\chi_{289}(45,\cdot)\)
\(\chi_{289}(46,\cdot)\)
\(\chi_{289}(48,\cdot)\)
\(\chi_{289}(54,\cdot)\)
\(\chi_{289}(56,\cdot)\)
\(\chi_{289}(57,\cdot)\)
\(\chi_{289}(58,\cdot)\)
\(\chi_{289}(61,\cdot)\)
\(\chi_{289}(62,\cdot)\)
\(\chi_{289}(63,\cdot)\)
...
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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 ]
\(3\) → \(e\left(\frac{135}{272}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 289 }(96, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{41}{136}\right)\) | \(e\left(\frac{135}{272}\right)\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{179}{272}\right)\) | \(e\left(\frac{217}{272}\right)\) | \(e\left(\frac{253}{272}\right)\) | \(e\left(\frac{123}{136}\right)\) | \(e\left(\frac{135}{136}\right)\) | \(e\left(\frac{261}{272}\right)\) | \(e\left(\frac{113}{272}\right)\) |
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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)
