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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([0,0,98,99]))
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pari:[g,chi] = znchar(Mod(13,8004))
\(\chi_{8004}(13,\cdot)\)
\(\chi_{8004}(121,\cdot)\)
\(\chi_{8004}(265,\cdot)\)
\(\chi_{8004}(325,\cdot)\)
\(\chi_{8004}(361,\cdot)\)
\(\chi_{8004}(469,\cdot)\)
\(\chi_{8004}(673,\cdot)\)
\(\chi_{8004}(817,\cdot)\)
\(\chi_{8004}(961,\cdot)\)
\(\chi_{8004}(1021,\cdot)\)
\(\chi_{8004}(1153,\cdot)\)
\(\chi_{8004}(1369,\cdot)\)
\(\chi_{8004}(1405,\cdot)\)
\(\chi_{8004}(1501,\cdot)\)
\(\chi_{8004}(1513,\cdot)\)
\(\chi_{8004}(1849,\cdot)\)
\(\chi_{8004}(1981,\cdot)\)
\(\chi_{8004}(2005,\cdot)\)
\(\chi_{8004}(2065,\cdot)\)
\(\chi_{8004}(2101,\cdot)\)
\(\chi_{8004}(2197,\cdot)\)
\(\chi_{8004}(2329,\cdot)\)
\(\chi_{8004}(2557,\cdot)\)
\(\chi_{8004}(2677,\cdot)\)
\(\chi_{8004}(2893,\cdot)\)
\(\chi_{8004}(3025,\cdot)\)
\(\chi_{8004}(3049,\cdot)\)
\(\chi_{8004}(3109,\cdot)\)
\(\chi_{8004}(3397,\cdot)\)
\(\chi_{8004}(3601,\cdot)\)
...
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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 ]
\((4003,2669,3133,553)\) → \((1,1,e\left(\frac{7}{11}\right),e\left(\frac{9}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 8004 }(13, a) \) |
\(1\) | \(1\) | \(e\left(\frac{60}{77}\right)\) | \(e\left(\frac{62}{77}\right)\) | \(e\left(\frac{123}{154}\right)\) | \(e\left(\frac{37}{77}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{51}{154}\right)\) | \(e\left(\frac{43}{77}\right)\) | \(e\left(\frac{71}{154}\right)\) | \(e\left(\frac{45}{77}\right)\) | \(e\left(\frac{45}{154}\right)\) |
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sage:chi.jacobi_sum(n)
