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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8004, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,0,24,33]))
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pari:[g,chi] = znchar(Mod(4135,8004))
\(\chi_{8004}(307,\cdot)\)
\(\chi_{8004}(331,\cdot)\)
\(\chi_{8004}(679,\cdot)\)
\(\chi_{8004}(1375,\cdot)\)
\(\chi_{8004}(2395,\cdot)\)
\(\chi_{8004}(2419,\cdot)\)
\(\chi_{8004}(2743,\cdot)\)
\(\chi_{8004}(3091,\cdot)\)
\(\chi_{8004}(3439,\cdot)\)
\(\chi_{8004}(3463,\cdot)\)
\(\chi_{8004}(3811,\cdot)\)
\(\chi_{8004}(4135,\cdot)\)
\(\chi_{8004}(4855,\cdot)\)
\(\chi_{8004}(5179,\cdot)\)
\(\chi_{8004}(5551,\cdot)\)
\(\chi_{8004}(6223,\cdot)\)
\(\chi_{8004}(6571,\cdot)\)
\(\chi_{8004}(7615,\cdot)\)
\(\chi_{8004}(7639,\cdot)\)
\(\chi_{8004}(7987,\cdot)\)
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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 ]
\((4003,2669,3133,553)\) → \((-1,1,e\left(\frac{6}{11}\right),-i)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 8004 }(4135, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{31}{44}\right)\) |
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sage:chi.jacobi_sum(n)
