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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7260, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,0,33,10]))
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pari:[g,chi] = znchar(Mod(43,7260))
\(\chi_{7260}(43,\cdot)\)
\(\chi_{7260}(307,\cdot)\)
\(\chi_{7260}(703,\cdot)\)
\(\chi_{7260}(1363,\cdot)\)
\(\chi_{7260}(1627,\cdot)\)
\(\chi_{7260}(2023,\cdot)\)
\(\chi_{7260}(2287,\cdot)\)
\(\chi_{7260}(2683,\cdot)\)
\(\chi_{7260}(2947,\cdot)\)
\(\chi_{7260}(3343,\cdot)\)
\(\chi_{7260}(3607,\cdot)\)
\(\chi_{7260}(4003,\cdot)\)
\(\chi_{7260}(4267,\cdot)\)
\(\chi_{7260}(4663,\cdot)\)
\(\chi_{7260}(4927,\cdot)\)
\(\chi_{7260}(5587,\cdot)\)
\(\chi_{7260}(5983,\cdot)\)
\(\chi_{7260}(6247,\cdot)\)
\(\chi_{7260}(6643,\cdot)\)
\(\chi_{7260}(6907,\cdot)\)
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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 ]
\((3631,4841,4357,7141)\) → \((-1,1,-i,e\left(\frac{5}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 7260 }(43, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{19}{44}\right)\) |
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sage:chi.jacobi_sum(n)
