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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1740, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,14,21,16]))
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pari:[g,chi] = znchar(Mod(953,1740))
\(\chi_{1740}(53,\cdot)\)
\(\chi_{1740}(197,\cdot)\)
\(\chi_{1740}(257,\cdot)\)
\(\chi_{1740}(413,\cdot)\)
\(\chi_{1740}(857,\cdot)\)
\(\chi_{1740}(893,\cdot)\)
\(\chi_{1740}(953,\cdot)\)
\(\chi_{1740}(977,\cdot)\)
\(\chi_{1740}(1097,\cdot)\)
\(\chi_{1740}(1457,\cdot)\)
\(\chi_{1740}(1553,\cdot)\)
\(\chi_{1740}(1673,\cdot)\)
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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 ]
\((871,581,697,901)\) → \((1,-1,-i,e\left(\frac{4}{7}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 1740 }(953, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(i\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(-1\) | \(e\left(\frac{19}{28}\right)\) |
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sage:chi.jacobi_sum(n)
