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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(648, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([0,0,53]))
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pari:[g,chi] = znchar(Mod(41,648))
\(\chi_{648}(41,\cdot)\)
\(\chi_{648}(65,\cdot)\)
\(\chi_{648}(113,\cdot)\)
\(\chi_{648}(137,\cdot)\)
\(\chi_{648}(185,\cdot)\)
\(\chi_{648}(209,\cdot)\)
\(\chi_{648}(257,\cdot)\)
\(\chi_{648}(281,\cdot)\)
\(\chi_{648}(329,\cdot)\)
\(\chi_{648}(353,\cdot)\)
\(\chi_{648}(401,\cdot)\)
\(\chi_{648}(425,\cdot)\)
\(\chi_{648}(473,\cdot)\)
\(\chi_{648}(497,\cdot)\)
\(\chi_{648}(545,\cdot)\)
\(\chi_{648}(569,\cdot)\)
\(\chi_{648}(617,\cdot)\)
\(\chi_{648}(641,\cdot)\)
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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 ]
\((487,325,569)\) → \((1,1,e\left(\frac{53}{54}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 648 }(41, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{17}{27}\right)\) |
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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)
