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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([0,22,14,33]))
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pari:[g,chi] = znchar(Mod(17,8004))
\(\chi_{8004}(17,\cdot)\)
\(\chi_{8004}(365,\cdot)\)
\(\chi_{8004}(389,\cdot)\)
\(\chi_{8004}(1433,\cdot)\)
\(\chi_{8004}(1781,\cdot)\)
\(\chi_{8004}(2453,\cdot)\)
\(\chi_{8004}(2825,\cdot)\)
\(\chi_{8004}(3149,\cdot)\)
\(\chi_{8004}(3869,\cdot)\)
\(\chi_{8004}(4193,\cdot)\)
\(\chi_{8004}(4541,\cdot)\)
\(\chi_{8004}(4565,\cdot)\)
\(\chi_{8004}(4913,\cdot)\)
\(\chi_{8004}(5261,\cdot)\)
\(\chi_{8004}(5585,\cdot)\)
\(\chi_{8004}(5609,\cdot)\)
\(\chi_{8004}(6629,\cdot)\)
\(\chi_{8004}(7325,\cdot)\)
\(\chi_{8004}(7673,\cdot)\)
\(\chi_{8004}(7697,\cdot)\)
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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 ]
\((4003,2669,3133,553)\) → \((1,-1,e\left(\frac{7}{22}\right),-i)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 8004 }(17, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{41}{44}\right)\) |
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sage:chi.jacobi_sum(n)
