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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(289, base_ring=CyclotomicField(136))
M = H._module
chi = DirichletCharacter(H, M([19]))
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pari:[g,chi] = znchar(Mod(49,289))
| Modulus: | \(289\) | |
| Conductor: | \(289\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(136\) |
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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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{289}(2,\cdot)\)
\(\chi_{289}(8,\cdot)\)
\(\chi_{289}(9,\cdot)\)
\(\chi_{289}(15,\cdot)\)
\(\chi_{289}(19,\cdot)\)
\(\chi_{289}(25,\cdot)\)
\(\chi_{289}(26,\cdot)\)
\(\chi_{289}(32,\cdot)\)
\(\chi_{289}(36,\cdot)\)
\(\chi_{289}(42,\cdot)\)
\(\chi_{289}(43,\cdot)\)
\(\chi_{289}(49,\cdot)\)
\(\chi_{289}(53,\cdot)\)
\(\chi_{289}(59,\cdot)\)
\(\chi_{289}(60,\cdot)\)
\(\chi_{289}(66,\cdot)\)
\(\chi_{289}(70,\cdot)\)
\(\chi_{289}(76,\cdot)\)
\(\chi_{289}(77,\cdot)\)
\(\chi_{289}(83,\cdot)\)
\(\chi_{289}(87,\cdot)\)
\(\chi_{289}(93,\cdot)\)
\(\chi_{289}(94,\cdot)\)
\(\chi_{289}(100,\cdot)\)
\(\chi_{289}(104,\cdot)\)
\(\chi_{289}(111,\cdot)\)
\(\chi_{289}(117,\cdot)\)
\(\chi_{289}(121,\cdot)\)
\(\chi_{289}(127,\cdot)\)
\(\chi_{289}(128,\cdot)\)
...
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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 ]
\(3\) → \(e\left(\frac{19}{136}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 289 }(49, a) \) |
\(1\) | \(1\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{19}{136}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{135}{136}\right)\) | \(e\left(\frac{93}{136}\right)\) | \(e\left(\frac{89}{136}\right)\) | \(e\left(\frac{43}{68}\right)\) | \(e\left(\frac{19}{68}\right)\) | \(e\left(\frac{73}{136}\right)\) | \(e\left(\frac{29}{136}\right)\) |
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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)
