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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3042, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([0,56]))
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pari:[g,chi] = znchar(Mod(55,3042))
\(\chi_{3042}(55,\cdot)\)
\(\chi_{3042}(217,\cdot)\)
\(\chi_{3042}(289,\cdot)\)
\(\chi_{3042}(451,\cdot)\)
\(\chi_{3042}(523,\cdot)\)
\(\chi_{3042}(685,\cdot)\)
\(\chi_{3042}(757,\cdot)\)
\(\chi_{3042}(919,\cdot)\)
\(\chi_{3042}(1153,\cdot)\)
\(\chi_{3042}(1225,\cdot)\)
\(\chi_{3042}(1387,\cdot)\)
\(\chi_{3042}(1459,\cdot)\)
\(\chi_{3042}(1621,\cdot)\)
\(\chi_{3042}(1693,\cdot)\)
\(\chi_{3042}(1855,\cdot)\)
\(\chi_{3042}(1927,\cdot)\)
\(\chi_{3042}(2089,\cdot)\)
\(\chi_{3042}(2161,\cdot)\)
\(\chi_{3042}(2323,\cdot)\)
\(\chi_{3042}(2395,\cdot)\)
\(\chi_{3042}(2629,\cdot)\)
\(\chi_{3042}(2791,\cdot)\)
\(\chi_{3042}(2863,\cdot)\)
\(\chi_{3042}(3025,\cdot)\)
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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 ]
\((677,847)\) → \((1,e\left(\frac{28}{39}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 3042 }(55, a) \) |
\(1\) | \(1\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{11}{39}\right)\) |
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sage:chi.jacobi_sum(n)
