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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8004, base_ring=CyclotomicField(154))
M = H._module
chi = DirichletCharacter(H, M([0,77,7,121]))
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pari:[g,chi] = znchar(Mod(5,8004))
\(\chi_{8004}(5,\cdot)\)
\(\chi_{8004}(125,\cdot)\)
\(\chi_{8004}(149,\cdot)\)
\(\chi_{8004}(245,\cdot)\)
\(\chi_{8004}(341,\cdot)\)
\(\chi_{8004}(497,\cdot)\)
\(\chi_{8004}(557,\cdot)\)
\(\chi_{8004}(701,\cdot)\)
\(\chi_{8004}(845,\cdot)\)
\(\chi_{8004}(941,\cdot)\)
\(\chi_{8004}(1049,\cdot)\)
\(\chi_{8004}(1169,\cdot)\)
\(\chi_{8004}(1193,\cdot)\)
\(\chi_{8004}(1253,\cdot)\)
\(\chi_{8004}(1385,\cdot)\)
\(\chi_{8004}(1397,\cdot)\)
\(\chi_{8004}(1601,\cdot)\)
\(\chi_{8004}(1745,\cdot)\)
\(\chi_{8004}(1949,\cdot)\)
\(\chi_{8004}(1985,\cdot)\)
\(\chi_{8004}(2081,\cdot)\)
\(\chi_{8004}(2213,\cdot)\)
\(\chi_{8004}(2297,\cdot)\)
\(\chi_{8004}(2333,\cdot)\)
\(\chi_{8004}(2429,\cdot)\)
\(\chi_{8004}(2777,\cdot)\)
\(\chi_{8004}(2909,\cdot)\)
\(\chi_{8004}(3125,\cdot)\)
\(\chi_{8004}(3257,\cdot)\)
\(\chi_{8004}(3281,\cdot)\)
...
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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 ]
\((4003,2669,3133,553)\) → \((1,-1,e\left(\frac{1}{22}\right),e\left(\frac{11}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 8004 }(5, a) \) |
\(1\) | \(1\) | \(e\left(\frac{64}{77}\right)\) | \(e\left(\frac{45}{154}\right)\) | \(e\left(\frac{85}{154}\right)\) | \(e\left(\frac{60}{77}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{58}{77}\right)\) | \(e\left(\frac{51}{77}\right)\) | \(e\left(\frac{9}{154}\right)\) | \(e\left(\frac{19}{154}\right)\) | \(e\left(\frac{24}{77}\right)\) |
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sage:chi.jacobi_sum(n)
