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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3042, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([78,71]))
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pari:[g,chi] = znchar(Mod(1385,3042))
\(\chi_{3042}(71,\cdot)\)
\(\chi_{3042}(197,\cdot)\)
\(\chi_{3042}(215,\cdot)\)
\(\chi_{3042}(305,\cdot)\)
\(\chi_{3042}(323,\cdot)\)
\(\chi_{3042}(431,\cdot)\)
\(\chi_{3042}(449,\cdot)\)
\(\chi_{3042}(539,\cdot)\)
\(\chi_{3042}(557,\cdot)\)
\(\chi_{3042}(665,\cdot)\)
\(\chi_{3042}(683,\cdot)\)
\(\chi_{3042}(773,\cdot)\)
\(\chi_{3042}(791,\cdot)\)
\(\chi_{3042}(899,\cdot)\)
\(\chi_{3042}(917,\cdot)\)
\(\chi_{3042}(1007,\cdot)\)
\(\chi_{3042}(1025,\cdot)\)
\(\chi_{3042}(1133,\cdot)\)
\(\chi_{3042}(1151,\cdot)\)
\(\chi_{3042}(1241,\cdot)\)
\(\chi_{3042}(1259,\cdot)\)
\(\chi_{3042}(1367,\cdot)\)
\(\chi_{3042}(1385,\cdot)\)
\(\chi_{3042}(1475,\cdot)\)
\(\chi_{3042}(1493,\cdot)\)
\(\chi_{3042}(1619,\cdot)\)
\(\chi_{3042}(1727,\cdot)\)
\(\chi_{3042}(1835,\cdot)\)
\(\chi_{3042}(1853,\cdot)\)
\(\chi_{3042}(1943,\cdot)\)
...
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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 ]
\((677,847)\) → \((-1,e\left(\frac{71}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 3042 }(1385, a) \) |
\(1\) | \(1\) | \(e\left(\frac{31}{52}\right)\) | \(e\left(\frac{109}{156}\right)\) | \(e\left(\frac{59}{156}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{29}{52}\right)\) | \(e\left(\frac{23}{78}\right)\) |
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sage:chi.jacobi_sum(n)
