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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2656, base_ring=CyclotomicField(82))
M = H._module
chi = DirichletCharacter(H, M([41,41,21]))
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pari:[g,chi] = znchar(Mod(655,2656))
\(\chi_{2656}(15,\cdot)\)
\(\chi_{2656}(47,\cdot)\)
\(\chi_{2656}(79,\cdot)\)
\(\chi_{2656}(143,\cdot)\)
\(\chi_{2656}(239,\cdot)\)
\(\chi_{2656}(271,\cdot)\)
\(\chi_{2656}(303,\cdot)\)
\(\chi_{2656}(367,\cdot)\)
\(\chi_{2656}(399,\cdot)\)
\(\chi_{2656}(495,\cdot)\)
\(\chi_{2656}(623,\cdot)\)
\(\chi_{2656}(655,\cdot)\)
\(\chi_{2656}(719,\cdot)\)
\(\chi_{2656}(975,\cdot)\)
\(\chi_{2656}(1039,\cdot)\)
\(\chi_{2656}(1103,\cdot)\)
\(\chi_{2656}(1135,\cdot)\)
\(\chi_{2656}(1167,\cdot)\)
\(\chi_{2656}(1263,\cdot)\)
\(\chi_{2656}(1295,\cdot)\)
\(\chi_{2656}(1487,\cdot)\)
\(\chi_{2656}(1551,\cdot)\)
\(\chi_{2656}(1583,\cdot)\)
\(\chi_{2656}(1679,\cdot)\)
\(\chi_{2656}(1775,\cdot)\)
\(\chi_{2656}(1839,\cdot)\)
\(\chi_{2656}(1871,\cdot)\)
\(\chi_{2656}(1967,\cdot)\)
\(\chi_{2656}(2031,\cdot)\)
\(\chi_{2656}(2063,\cdot)\)
...
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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 ]
\((831,997,417)\) → \((-1,-1,e\left(\frac{21}{82}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 2656 }(655, a) \) |
\(1\) | \(1\) | \(e\left(\frac{18}{41}\right)\) | \(e\left(\frac{17}{41}\right)\) | \(e\left(\frac{45}{82}\right)\) | \(e\left(\frac{36}{41}\right)\) | \(e\left(\frac{6}{41}\right)\) | \(e\left(\frac{9}{41}\right)\) | \(e\left(\frac{35}{41}\right)\) | \(e\left(\frac{14}{41}\right)\) | \(e\left(\frac{3}{82}\right)\) | \(e\left(\frac{81}{82}\right)\) |
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sage:chi.jacobi_sum(n)
