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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([154,0,28,187]))
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pari:[g,chi] = znchar(Mod(1819,8004))
\(\chi_{8004}(31,\cdot)\)
\(\chi_{8004}(55,\cdot)\)
\(\chi_{8004}(127,\cdot)\)
\(\chi_{8004}(163,\cdot)\)
\(\chi_{8004}(211,\cdot)\)
\(\chi_{8004}(259,\cdot)\)
\(\chi_{8004}(271,\cdot)\)
\(\chi_{8004}(403,\cdot)\)
\(\chi_{8004}(427,\cdot)\)
\(\chi_{8004}(583,\cdot)\)
\(\chi_{8004}(607,\cdot)\)
\(\chi_{8004}(715,\cdot)\)
\(\chi_{8004}(739,\cdot)\)
\(\chi_{8004}(775,\cdot)\)
\(\chi_{8004}(823,\cdot)\)
\(\chi_{8004}(859,\cdot)\)
\(\chi_{8004}(955,\cdot)\)
\(\chi_{8004}(1087,\cdot)\)
\(\chi_{8004}(1099,\cdot)\)
\(\chi_{8004}(1255,\cdot)\)
\(\chi_{8004}(1291,\cdot)\)
\(\chi_{8004}(1315,\cdot)\)
\(\chi_{8004}(1411,\cdot)\)
\(\chi_{8004}(1435,\cdot)\)
\(\chi_{8004}(1603,\cdot)\)
\(\chi_{8004}(1639,\cdot)\)
\(\chi_{8004}(1651,\cdot)\)
\(\chi_{8004}(1783,\cdot)\)
\(\chi_{8004}(1819,\cdot)\)
\(\chi_{8004}(1867,\cdot)\)
...
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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 ]
\((4003,2669,3133,553)\) → \((-1,1,e\left(\frac{1}{11}\right),e\left(\frac{17}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 8004 }(1819, a) \) |
\(1\) | \(1\) | \(e\left(\frac{69}{154}\right)\) | \(e\left(\frac{79}{154}\right)\) | \(e\left(\frac{153}{308}\right)\) | \(e\left(\frac{31}{154}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{101}{308}\right)\) | \(e\left(\frac{69}{77}\right)\) | \(e\left(\frac{201}{308}\right)\) | \(e\left(\frac{74}{77}\right)\) | \(e\left(\frac{225}{308}\right)\) |
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sage:chi.jacobi_sum(n)
