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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(309680, base_ring=CyclotomicField(52))
M = H._module
chi = DirichletCharacter(H, M([0,0,13,26,4]))
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pari:[g,chi] = znchar(Mod(97,309680))
\(\chi_{309680}(97,\cdot)\)
\(\chi_{309680}(4017,\cdot)\)
\(\chi_{309680}(14993,\cdot)\)
\(\chi_{309680}(30673,\cdot)\)
\(\chi_{309680}(35377,\cdot)\)
\(\chi_{309680}(42433,\cdot)\)
\(\chi_{309680}(62033,\cdot)\)
\(\chi_{309680}(65953,\cdot)\)
\(\chi_{309680}(97313,\cdot)\)
\(\chi_{309680}(160817,\cdot)\)
\(\chi_{309680}(164737,\cdot)\)
\(\chi_{309680}(203937,\cdot)\)
\(\chi_{309680}(207857,\cdot)\)
\(\chi_{309680}(222753,\cdot)\)
\(\chi_{309680}(223537,\cdot)\)
\(\chi_{309680}(226673,\cdot)\)
\(\chi_{309680}(231377,\cdot)\)
\(\chi_{309680}(262737,\cdot)\)
\(\chi_{309680}(265873,\cdot)\)
\(\chi_{309680}(269793,\cdot)\)
\(\chi_{309680}(278417,\cdot)\)
\(\chi_{309680}(285473,\cdot)\)
\(\chi_{309680}(290177,\cdot)\)
\(\chi_{309680}(293313,\cdot)\)
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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 ]
\((193551,232261,61937,297041,82321)\) → \((1,1,i,-1,e\left(\frac{1}{13}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 309680 }(97, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{52}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{45}{52}\right)\) | \(e\left(\frac{19}{52}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(-i\) | \(e\left(\frac{51}{52}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{21}{26}\right)\) |
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sage:chi.jacobi_sum(n)
