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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([0,7]))
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pari:[g,chi] = znchar(Mod(31,1014))
\(\chi_{1014}(31,\cdot)\)
\(\chi_{1014}(73,\cdot)\)
\(\chi_{1014}(109,\cdot)\)
\(\chi_{1014}(151,\cdot)\)
\(\chi_{1014}(187,\cdot)\)
\(\chi_{1014}(229,\cdot)\)
\(\chi_{1014}(265,\cdot)\)
\(\chi_{1014}(307,\cdot)\)
\(\chi_{1014}(343,\cdot)\)
\(\chi_{1014}(385,\cdot)\)
\(\chi_{1014}(421,\cdot)\)
\(\chi_{1014}(463,\cdot)\)
\(\chi_{1014}(499,\cdot)\)
\(\chi_{1014}(541,\cdot)\)
\(\chi_{1014}(619,\cdot)\)
\(\chi_{1014}(655,\cdot)\)
\(\chi_{1014}(697,\cdot)\)
\(\chi_{1014}(733,\cdot)\)
\(\chi_{1014}(811,\cdot)\)
\(\chi_{1014}(853,\cdot)\)
\(\chi_{1014}(889,\cdot)\)
\(\chi_{1014}(931,\cdot)\)
\(\chi_{1014}(967,\cdot)\)
\(\chi_{1014}(1009,\cdot)\)
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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 ]
\((677,847)\) → \((1,e\left(\frac{7}{52}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 1014 }(31, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{11}{52}\right)\) | \(e\left(\frac{21}{52}\right)\) | \(e\left(\frac{45}{52}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(-i\) | \(-1\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{43}{52}\right)\) | \(e\left(\frac{8}{13}\right)\) |
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sage:chi.jacobi_sum(n)
