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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(40310, base_ring=CyclotomicField(1932))
M = H._module
chi = DirichletCharacter(H, M([483,828,700]))
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pari:[g,chi] = znchar(Mod(7,40310))
\(\chi_{40310}(7,\cdot)\)
\(\chi_{40310}(83,\cdot)\)
\(\chi_{40310}(107,\cdot)\)
\(\chi_{40310}(257,\cdot)\)
\(\chi_{40310}(313,\cdot)\)
\(\chi_{40310}(567,\cdot)\)
\(\chi_{40310}(587,\cdot)\)
\(\chi_{40310}(603,\cdot)\)
\(\chi_{40310}(663,\cdot)\)
\(\chi_{40310}(683,\cdot)\)
\(\chi_{40310}(923,\cdot)\)
\(\chi_{40310}(977,\cdot)\)
\(\chi_{40310}(993,\cdot)\)
\(\chi_{40310}(1093,\cdot)\)
\(\chi_{40310}(1097,\cdot)\)
\(\chi_{40310}(1147,\cdot)\)
\(\chi_{40310}(1183,\cdot)\)
\(\chi_{40310}(1267,\cdot)\)
\(\chi_{40310}(1387,\cdot)\)
\(\chi_{40310}(1437,\cdot)\)
\(\chi_{40310}(1457,\cdot)\)
\(\chi_{40310}(1473,\cdot)\)
\(\chi_{40310}(1503,\cdot)\)
\(\chi_{40310}(1533,\cdot)\)
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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 ]
\((24187,19461,16821)\) → \((i,e\left(\frac{3}{7}\right),e\left(\frac{25}{69}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 40310 }(7, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1445}{1932}\right)\) | \(e\left(\frac{983}{1932}\right)\) | \(e\left(\frac{479}{966}\right)\) | \(e\left(\frac{121}{483}\right)\) | \(e\left(\frac{1261}{1932}\right)\) | \(e\left(\frac{5}{276}\right)\) | \(e\left(\frac{443}{966}\right)\) | \(e\left(\frac{124}{483}\right)\) | \(e\left(\frac{67}{644}\right)\) | \(e\left(\frac{157}{644}\right)\) |
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sage:chi.jacobi_sum(n)
