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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([154,0,98,55]))
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pari:[g,chi] = znchar(Mod(1627,8004))
\(\chi_{8004}(19,\cdot)\)
\(\chi_{8004}(43,\cdot)\)
\(\chi_{8004}(79,\cdot)\)
\(\chi_{8004}(235,\cdot)\)
\(\chi_{8004}(247,\cdot)\)
\(\chi_{8004}(379,\cdot)\)
\(\chi_{8004}(475,\cdot)\)
\(\chi_{8004}(511,\cdot)\)
\(\chi_{8004}(559,\cdot)\)
\(\chi_{8004}(595,\cdot)\)
\(\chi_{8004}(619,\cdot)\)
\(\chi_{8004}(727,\cdot)\)
\(\chi_{8004}(751,\cdot)\)
\(\chi_{8004}(907,\cdot)\)
\(\chi_{8004}(931,\cdot)\)
\(\chi_{8004}(1063,\cdot)\)
\(\chi_{8004}(1075,\cdot)\)
\(\chi_{8004}(1123,\cdot)\)
\(\chi_{8004}(1171,\cdot)\)
\(\chi_{8004}(1207,\cdot)\)
\(\chi_{8004}(1279,\cdot)\)
\(\chi_{8004}(1303,\cdot)\)
\(\chi_{8004}(1423,\cdot)\)
\(\chi_{8004}(1447,\cdot)\)
\(\chi_{8004}(1555,\cdot)\)
\(\chi_{8004}(1627,\cdot)\)
\(\chi_{8004}(1663,\cdot)\)
\(\chi_{8004}(1759,\cdot)\)
\(\chi_{8004}(1903,\cdot)\)
\(\chi_{8004}(1951,\cdot)\)
...
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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 ]
\((4003,2669,3133,553)\) → \((-1,1,e\left(\frac{7}{22}\right),e\left(\frac{5}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 8004 }(1627, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{19}{77}\right)\) | \(e\left(\frac{53}{77}\right)\) | \(e\left(\frac{255}{308}\right)\) | \(e\left(\frac{103}{154}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{271}{308}\right)\) | \(e\left(\frac{38}{77}\right)\) | \(e\left(\frac{181}{308}\right)\) | \(e\left(\frac{72}{77}\right)\) | \(e\left(\frac{67}{308}\right)\) |
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sage:chi.jacobi_sum(n)
