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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([0,154,224,55]))
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pari:[g,chi] = znchar(Mod(1337,8004))
\(\chi_{8004}(77,\cdot)\)
\(\chi_{8004}(101,\cdot)\)
\(\chi_{8004}(269,\cdot)\)
\(\chi_{8004}(305,\cdot)\)
\(\chi_{8004}(317,\cdot)\)
\(\chi_{8004}(449,\cdot)\)
\(\chi_{8004}(485,\cdot)\)
\(\chi_{8004}(533,\cdot)\)
\(\chi_{8004}(653,\cdot)\)
\(\chi_{8004}(785,\cdot)\)
\(\chi_{8004}(809,\cdot)\)
\(\chi_{8004}(1001,\cdot)\)
\(\chi_{8004}(1025,\cdot)\)
\(\chi_{8004}(1133,\cdot)\)
\(\chi_{8004}(1145,\cdot)\)
\(\chi_{8004}(1181,\cdot)\)
\(\chi_{8004}(1313,\cdot)\)
\(\chi_{8004}(1337,\cdot)\)
\(\chi_{8004}(1361,\cdot)\)
\(\chi_{8004}(1373,\cdot)\)
\(\chi_{8004}(1481,\cdot)\)
\(\chi_{8004}(1577,\cdot)\)
\(\chi_{8004}(1613,\cdot)\)
\(\chi_{8004}(1685,\cdot)\)
\(\chi_{8004}(1697,\cdot)\)
\(\chi_{8004}(1829,\cdot)\)
\(\chi_{8004}(1853,\cdot)\)
\(\chi_{8004}(1925,\cdot)\)
\(\chi_{8004}(1961,\cdot)\)
\(\chi_{8004}(2009,\cdot)\)
...
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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 ]
\((4003,2669,3133,553)\) → \((1,-1,e\left(\frac{8}{11}\right),e\left(\frac{5}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 8004 }(1337, a) \) |
\(1\) | \(1\) | \(e\left(\frac{12}{77}\right)\) | \(e\left(\frac{74}{77}\right)\) | \(e\left(\frac{157}{308}\right)\) | \(e\left(\frac{61}{154}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{159}{308}\right)\) | \(e\left(\frac{24}{77}\right)\) | \(e\left(\frac{167}{308}\right)\) | \(e\left(\frac{9}{77}\right)\) | \(e\left(\frac{249}{308}\right)\) |
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sage:chi.jacobi_sum(n)
