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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1014, base_ring=CyclotomicField(52))
M = H._module
chi = DirichletCharacter(H, M([26,51]))
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pari:[g,chi] = znchar(Mod(317,1014))
\(\chi_{1014}(5,\cdot)\)
\(\chi_{1014}(47,\cdot)\)
\(\chi_{1014}(83,\cdot)\)
\(\chi_{1014}(125,\cdot)\)
\(\chi_{1014}(161,\cdot)\)
\(\chi_{1014}(203,\cdot)\)
\(\chi_{1014}(281,\cdot)\)
\(\chi_{1014}(317,\cdot)\)
\(\chi_{1014}(359,\cdot)\)
\(\chi_{1014}(395,\cdot)\)
\(\chi_{1014}(473,\cdot)\)
\(\chi_{1014}(515,\cdot)\)
\(\chi_{1014}(551,\cdot)\)
\(\chi_{1014}(593,\cdot)\)
\(\chi_{1014}(629,\cdot)\)
\(\chi_{1014}(671,\cdot)\)
\(\chi_{1014}(707,\cdot)\)
\(\chi_{1014}(749,\cdot)\)
\(\chi_{1014}(785,\cdot)\)
\(\chi_{1014}(827,\cdot)\)
\(\chi_{1014}(863,\cdot)\)
\(\chi_{1014}(905,\cdot)\)
\(\chi_{1014}(941,\cdot)\)
\(\chi_{1014}(983,\cdot)\)
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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 ]
\((677,847)\) → \((-1,e\left(\frac{51}{52}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 1014 }(317, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{52}\right)\) | \(e\left(\frac{49}{52}\right)\) | \(e\left(\frac{27}{52}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(-i\) | \(1\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{31}{52}\right)\) | \(e\left(\frac{7}{26}\right)\) |
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sage:chi.jacobi_sum(n)
