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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9800, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([42,42,63,32]))
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pari:[g,chi] = znchar(Mod(4043,9800))
\(\chi_{9800}(107,\cdot)\)
\(\chi_{9800}(443,\cdot)\)
\(\chi_{9800}(907,\cdot)\)
\(\chi_{9800}(1507,\cdot)\)
\(\chi_{9800}(2307,\cdot)\)
\(\chi_{9800}(2643,\cdot)\)
\(\chi_{9800}(2907,\cdot)\)
\(\chi_{9800}(3243,\cdot)\)
\(\chi_{9800}(3707,\cdot)\)
\(\chi_{9800}(4043,\cdot)\)
\(\chi_{9800}(4307,\cdot)\)
\(\chi_{9800}(4643,\cdot)\)
\(\chi_{9800}(5107,\cdot)\)
\(\chi_{9800}(5443,\cdot)\)
\(\chi_{9800}(5707,\cdot)\)
\(\chi_{9800}(6043,\cdot)\)
\(\chi_{9800}(6507,\cdot)\)
\(\chi_{9800}(6843,\cdot)\)
\(\chi_{9800}(7107,\cdot)\)
\(\chi_{9800}(7443,\cdot)\)
\(\chi_{9800}(8243,\cdot)\)
\(\chi_{9800}(8843,\cdot)\)
\(\chi_{9800}(9307,\cdot)\)
\(\chi_{9800}(9643,\cdot)\)
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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 ]
\((7351,4901,1177,5001)\) → \((-1,-1,-i,e\left(\frac{8}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 9800 }(4043, a) \) |
\(1\) | \(1\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{6}\right)\) |
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sage:chi.jacobi_sum(n)
