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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([0,28,21,10]))
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pari:[g,chi] = znchar(Mod(28957,44100))
\(\chi_{44100}(493,\cdot)\)
\(\chi_{44100}(1993,\cdot)\)
\(\chi_{44100}(2257,\cdot)\)
\(\chi_{44100}(3757,\cdot)\)
\(\chi_{44100}(8293,\cdot)\)
\(\chi_{44100}(10057,\cdot)\)
\(\chi_{44100}(13093,\cdot)\)
\(\chi_{44100}(14593,\cdot)\)
\(\chi_{44100}(14857,\cdot)\)
\(\chi_{44100}(16357,\cdot)\)
\(\chi_{44100}(19393,\cdot)\)
\(\chi_{44100}(21157,\cdot)\)
\(\chi_{44100}(25693,\cdot)\)
\(\chi_{44100}(27193,\cdot)\)
\(\chi_{44100}(27457,\cdot)\)
\(\chi_{44100}(28957,\cdot)\)
\(\chi_{44100}(31993,\cdot)\)
\(\chi_{44100}(33493,\cdot)\)
\(\chi_{44100}(33757,\cdot)\)
\(\chi_{44100}(35257,\cdot)\)
\(\chi_{44100}(38293,\cdot)\)
\(\chi_{44100}(39793,\cdot)\)
\(\chi_{44100}(40057,\cdot)\)
\(\chi_{44100}(41557,\cdot)\)
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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 ]
\((22051,34301,15877,9901)\) → \((1,e\left(\frac{1}{3}\right),i,e\left(\frac{5}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 44100 }(28957, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(-1\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{67}{84}\right)\) |
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sage:chi.jacobi_sum(n)
