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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([42,0,21,16]))
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pari:[g,chi] = znchar(Mod(43507,44100))
\(\chi_{44100}(3943,\cdot)\)
\(\chi_{44100}(5707,\cdot)\)
\(\chi_{44100}(7543,\cdot)\)
\(\chi_{44100}(9307,\cdot)\)
\(\chi_{44100}(10243,\cdot)\)
\(\chi_{44100}(12007,\cdot)\)
\(\chi_{44100}(13843,\cdot)\)
\(\chi_{44100}(15607,\cdot)\)
\(\chi_{44100}(20143,\cdot)\)
\(\chi_{44100}(21907,\cdot)\)
\(\chi_{44100}(22843,\cdot)\)
\(\chi_{44100}(24607,\cdot)\)
\(\chi_{44100}(26443,\cdot)\)
\(\chi_{44100}(28207,\cdot)\)
\(\chi_{44100}(29143,\cdot)\)
\(\chi_{44100}(30907,\cdot)\)
\(\chi_{44100}(32743,\cdot)\)
\(\chi_{44100}(34507,\cdot)\)
\(\chi_{44100}(35443,\cdot)\)
\(\chi_{44100}(37207,\cdot)\)
\(\chi_{44100}(39043,\cdot)\)
\(\chi_{44100}(40807,\cdot)\)
\(\chi_{44100}(41743,\cdot)\)
\(\chi_{44100}(43507,\cdot)\)
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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 ]
\((22051,34301,15877,9901)\) → \((-1,1,i,e\left(\frac{4}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 44100 }(43507, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{11}{28}\right)\) |
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sage:chi.jacobi_sum(n)
