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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([0,77,105,44]))
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pari:[g,chi] = znchar(Mod(1445,8004))
\(\chi_{8004}(53,\cdot)\)
\(\chi_{8004}(65,\cdot)\)
\(\chi_{8004}(281,\cdot)\)
\(\chi_{8004}(401,\cdot)\)
\(\chi_{8004}(605,\cdot)\)
\(\chi_{8004}(893,\cdot)\)
\(\chi_{8004}(953,\cdot)\)
\(\chi_{8004}(977,\cdot)\)
\(\chi_{8004}(1109,\cdot)\)
\(\chi_{8004}(1325,\cdot)\)
\(\chi_{8004}(1445,\cdot)\)
\(\chi_{8004}(1673,\cdot)\)
\(\chi_{8004}(1805,\cdot)\)
\(\chi_{8004}(1901,\cdot)\)
\(\chi_{8004}(1937,\cdot)\)
\(\chi_{8004}(1997,\cdot)\)
\(\chi_{8004}(2021,\cdot)\)
\(\chi_{8004}(2153,\cdot)\)
\(\chi_{8004}(2489,\cdot)\)
\(\chi_{8004}(2501,\cdot)\)
\(\chi_{8004}(2597,\cdot)\)
\(\chi_{8004}(2633,\cdot)\)
\(\chi_{8004}(2849,\cdot)\)
\(\chi_{8004}(2981,\cdot)\)
\(\chi_{8004}(3041,\cdot)\)
\(\chi_{8004}(3185,\cdot)\)
\(\chi_{8004}(3329,\cdot)\)
\(\chi_{8004}(3533,\cdot)\)
\(\chi_{8004}(3641,\cdot)\)
\(\chi_{8004}(3677,\cdot)\)
...
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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 ]
\((4003,2669,3133,553)\) → \((1,-1,e\left(\frac{15}{22}\right),e\left(\frac{2}{7}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 8004 }(1445, a) \) |
\(1\) | \(1\) | \(e\left(\frac{36}{77}\right)\) | \(e\left(\frac{59}{154}\right)\) | \(e\left(\frac{60}{77}\right)\) | \(e\left(\frac{53}{77}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{123}{154}\right)\) | \(e\left(\frac{72}{77}\right)\) | \(e\left(\frac{29}{77}\right)\) | \(e\left(\frac{131}{154}\right)\) | \(e\left(\frac{27}{154}\right)\) |
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sage:chi.jacobi_sum(n)
