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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(44100, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,21,0,19]))
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pari:[g,chi] = znchar(Mod(4301,44100))
\(\chi_{44100}(1601,\cdot)\)
\(\chi_{44100}(4301,\cdot)\)
\(\chi_{44100}(7901,\cdot)\)
\(\chi_{44100}(10601,\cdot)\)
\(\chi_{44100}(14201,\cdot)\)
\(\chi_{44100}(16901,\cdot)\)
\(\chi_{44100}(23201,\cdot)\)
\(\chi_{44100}(26801,\cdot)\)
\(\chi_{44100}(29501,\cdot)\)
\(\chi_{44100}(33101,\cdot)\)
\(\chi_{44100}(39401,\cdot)\)
\(\chi_{44100}(42101,\cdot)\)
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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 ]
\((22051,34301,15877,9901)\) → \((1,-1,1,e\left(\frac{19}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 44100 }(4301, a) \) |
\(1\) | \(1\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) |
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sage:chi.jacobi_sum(n)
