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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(40310, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([23,0,7]))
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pari:[g,chi] = znchar(Mod(59,40310))
\(\chi_{40310}(59,\cdot)\)
\(\chi_{40310}(929,\cdot)\)
\(\chi_{40310}(3829,\cdot)\)
\(\chi_{40310}(6149,\cdot)\)
\(\chi_{40310}(7889,\cdot)\)
\(\chi_{40310}(9049,\cdot)\)
\(\chi_{40310}(10209,\cdot)\)
\(\chi_{40310}(10499,\cdot)\)
\(\chi_{40310}(14559,\cdot)\)
\(\chi_{40310}(16589,\cdot)\)
\(\chi_{40310}(16879,\cdot)\)
\(\chi_{40310}(17459,\cdot)\)
\(\chi_{40310}(22389,\cdot)\)
\(\chi_{40310}(24129,\cdot)\)
\(\chi_{40310}(24419,\cdot)\)
\(\chi_{40310}(26159,\cdot)\)
\(\chi_{40310}(26449,\cdot)\)
\(\chi_{40310}(27319,\cdot)\)
\(\chi_{40310}(27609,\cdot)\)
\(\chi_{40310}(29059,\cdot)\)
\(\chi_{40310}(35439,\cdot)\)
\(\chi_{40310}(39499,\cdot)\)
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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 ]
\((24187,19461,16821)\) → \((-1,1,e\left(\frac{7}{46}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 40310 }(59, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{5}{46}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{11}{46}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{13}{46}\right)\) | \(e\left(\frac{39}{46}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) |
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sage:chi.jacobi_sum(n)
