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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3234, base_ring=CyclotomicField(70))
M = H._module
chi = DirichletCharacter(H, M([35,45,56]))
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pari:[g,chi] = znchar(Mod(839,3234))
\(\chi_{3234}(125,\cdot)\)
\(\chi_{3234}(251,\cdot)\)
\(\chi_{3234}(335,\cdot)\)
\(\chi_{3234}(377,\cdot)\)
\(\chi_{3234}(713,\cdot)\)
\(\chi_{3234}(797,\cdot)\)
\(\chi_{3234}(839,\cdot)\)
\(\chi_{3234}(1049,\cdot)\)
\(\chi_{3234}(1259,\cdot)\)
\(\chi_{3234}(1301,\cdot)\)
\(\chi_{3234}(1511,\cdot)\)
\(\chi_{3234}(1637,\cdot)\)
\(\chi_{3234}(1721,\cdot)\)
\(\chi_{3234}(1973,\cdot)\)
\(\chi_{3234}(2099,\cdot)\)
\(\chi_{3234}(2183,\cdot)\)
\(\chi_{3234}(2225,\cdot)\)
\(\chi_{3234}(2435,\cdot)\)
\(\chi_{3234}(2561,\cdot)\)
\(\chi_{3234}(2687,\cdot)\)
\(\chi_{3234}(2897,\cdot)\)
\(\chi_{3234}(3023,\cdot)\)
\(\chi_{3234}(3107,\cdot)\)
\(\chi_{3234}(3149,\cdot)\)
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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 ]
\((1079,199,2059)\) → \((-1,e\left(\frac{9}{14}\right),e\left(\frac{4}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3234 }(839, a) \) |
\(1\) | \(1\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{1}{70}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{47}{70}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) |
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sage:chi.jacobi_sum(n)
