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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(44100, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,15,21,20]))
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pari:[g,chi] = znchar(Mod(10259,44100))
\(\chi_{44100}(1439,\cdot)\)
\(\chi_{44100}(7919,\cdot)\)
\(\chi_{44100}(10259,\cdot)\)
\(\chi_{44100}(16739,\cdot)\)
\(\chi_{44100}(19079,\cdot)\)
\(\chi_{44100}(25559,\cdot)\)
\(\chi_{44100}(34379,\cdot)\)
\(\chi_{44100}(36719,\cdot)\)
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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 ]
\((22051,34301,15877,9901)\) → \((-1,-1,e\left(\frac{7}{10}\right),e\left(\frac{2}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 44100 }(10259, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(1\) |
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sage:chi.jacobi_sum(n)
