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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8004, base_ring=CyclotomicField(308))
M = H._module
chi = DirichletCharacter(H, M([0,0,210,143]))
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pari:[g,chi] = znchar(Mod(1261,8004))
\(\chi_{8004}(37,\cdot)\)
\(\chi_{8004}(61,\cdot)\)
\(\chi_{8004}(97,\cdot)\)
\(\chi_{8004}(205,\cdot)\)
\(\chi_{8004}(217,\cdot)\)
\(\chi_{8004}(337,\cdot)\)
\(\chi_{8004}(385,\cdot)\)
\(\chi_{8004}(421,\cdot)\)
\(\chi_{8004}(433,\cdot)\)
\(\chi_{8004}(649,\cdot)\)
\(\chi_{8004}(733,\cdot)\)
\(\chi_{8004}(757,\cdot)\)
\(\chi_{8004}(769,\cdot)\)
\(\chi_{8004}(793,\cdot)\)
\(\chi_{8004}(889,\cdot)\)
\(\chi_{8004}(925,\cdot)\)
\(\chi_{8004}(1033,\cdot)\)
\(\chi_{8004}(1141,\cdot)\)
\(\chi_{8004}(1249,\cdot)\)
\(\chi_{8004}(1261,\cdot)\)
\(\chi_{8004}(1345,\cdot)\)
\(\chi_{8004}(1477,\cdot)\)
\(\chi_{8004}(1489,\cdot)\)
\(\chi_{8004}(1585,\cdot)\)
\(\chi_{8004}(1597,\cdot)\)
\(\chi_{8004}(1621,\cdot)\)
\(\chi_{8004}(1693,\cdot)\)
\(\chi_{8004}(1801,\cdot)\)
\(\chi_{8004}(1813,\cdot)\)
\(\chi_{8004}(1837,\cdot)\)
...
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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 ]
\((4003,2669,3133,553)\) → \((1,1,e\left(\frac{15}{22}\right),e\left(\frac{13}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 8004 }(1261, a) \) |
\(1\) | \(1\) | \(e\left(\frac{69}{77}\right)\) | \(e\left(\frac{81}{154}\right)\) | \(e\left(\frac{229}{308}\right)\) | \(e\left(\frac{139}{154}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{125}{308}\right)\) | \(e\left(\frac{61}{77}\right)\) | \(e\left(\frac{171}{308}\right)\) | \(e\left(\frac{65}{154}\right)\) | \(e\left(\frac{219}{308}\right)\) |
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sage:chi.jacobi_sum(n)
