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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(44100, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([42,28,63,32]))
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pari:[g,chi] = znchar(Mod(28543,44100))
\(\chi_{44100}(3343,\cdot)\)
\(\chi_{44100}(5107,\cdot)\)
\(\chi_{44100}(8143,\cdot)\)
\(\chi_{44100}(9643,\cdot)\)
\(\chi_{44100}(9907,\cdot)\)
\(\chi_{44100}(11407,\cdot)\)
\(\chi_{44100}(14443,\cdot)\)
\(\chi_{44100}(16207,\cdot)\)
\(\chi_{44100}(20743,\cdot)\)
\(\chi_{44100}(22243,\cdot)\)
\(\chi_{44100}(22507,\cdot)\)
\(\chi_{44100}(24007,\cdot)\)
\(\chi_{44100}(27043,\cdot)\)
\(\chi_{44100}(28543,\cdot)\)
\(\chi_{44100}(28807,\cdot)\)
\(\chi_{44100}(30307,\cdot)\)
\(\chi_{44100}(33343,\cdot)\)
\(\chi_{44100}(34843,\cdot)\)
\(\chi_{44100}(35107,\cdot)\)
\(\chi_{44100}(36607,\cdot)\)
\(\chi_{44100}(39643,\cdot)\)
\(\chi_{44100}(41143,\cdot)\)
\(\chi_{44100}(41407,\cdot)\)
\(\chi_{44100}(42907,\cdot)\)
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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 ]
\((22051,34301,15877,9901)\) → \((-1,e\left(\frac{1}{3}\right),-i,e\left(\frac{8}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 44100 }(28543, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{31}{84}\right)\) |
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sage:chi.jacobi_sum(n)
