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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(805, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,55,54]))
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pari:[g,chi] = znchar(Mod(236,805))
\(\chi_{805}(26,\cdot)\)
\(\chi_{805}(31,\cdot)\)
\(\chi_{805}(96,\cdot)\)
\(\chi_{805}(101,\cdot)\)
\(\chi_{805}(131,\cdot)\)
\(\chi_{805}(236,\cdot)\)
\(\chi_{805}(271,\cdot)\)
\(\chi_{805}(311,\cdot)\)
\(\chi_{805}(376,\cdot)\)
\(\chi_{805}(381,\cdot)\)
\(\chi_{805}(416,\cdot)\)
\(\chi_{805}(446,\cdot)\)
\(\chi_{805}(486,\cdot)\)
\(\chi_{805}(556,\cdot)\)
\(\chi_{805}(591,\cdot)\)
\(\chi_{805}(656,\cdot)\)
\(\chi_{805}(696,\cdot)\)
\(\chi_{805}(726,\cdot)\)
\(\chi_{805}(731,\cdot)\)
\(\chi_{805}(761,\cdot)\)
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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 ]
\((162,346,281)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{9}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 805 }(236, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{7}{33}\right)\) |
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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)
