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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(309680, base_ring=CyclotomicField(1092))
M = H._module
chi = DirichletCharacter(H, M([0,0,273,650,294]))
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pari:[g,chi] = znchar(Mod(17,309680))
\(\chi_{309680}(17,\cdot)\)
\(\chi_{309680}(33,\cdot)\)
\(\chi_{309680}(817,\cdot)\)
\(\chi_{309680}(1937,\cdot)\)
\(\chi_{309680}(2033,\cdot)\)
\(\chi_{309680}(4177,\cdot)\)
\(\chi_{309680}(5073,\cdot)\)
\(\chi_{309680}(6177,\cdot)\)
\(\chi_{309680}(10657,\cdot)\)
\(\chi_{309680}(10993,\cdot)\)
\(\chi_{309680}(11233,\cdot)\)
\(\chi_{309680}(14577,\cdot)\)
\(\chi_{309680}(14593,\cdot)\)
\(\chi_{309680}(14913,\cdot)\)
\(\chi_{309680}(15713,\cdot)\)
\(\chi_{309680}(15937,\cdot)\)
\(\chi_{309680}(16817,\cdot)\)
\(\chi_{309680}(17713,\cdot)\)
\(\chi_{309680}(18513,\cdot)\)
\(\chi_{309680}(19633,\cdot)\)
\(\chi_{309680}(23873,\cdot)\)
\(\chi_{309680}(24897,\cdot)\)
\(\chi_{309680}(27233,\cdot)\)
\(\chi_{309680}(28577,\cdot)\)
\(\chi_{309680}(28817,\cdot)\)
\(\chi_{309680}(31153,\cdot)\)
\(\chi_{309680}(31377,\cdot)\)
\(\chi_{309680}(31617,\cdot)\)
\(\chi_{309680}(34513,\cdot)\)
\(\chi_{309680}(36433,\cdot)\)
...
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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 ]
\((193551,232261,61937,297041,82321)\) → \((1,1,i,e\left(\frac{25}{42}\right),e\left(\frac{7}{26}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 309680 }(17, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{671}{1092}\right)\) | \(e\left(\frac{125}{546}\right)\) | \(e\left(\frac{32}{273}\right)\) | \(e\left(\frac{199}{364}\right)\) | \(e\left(\frac{857}{1092}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{307}{364}\right)\) | \(e\left(\frac{16}{91}\right)\) | \(e\left(\frac{19}{78}\right)\) |
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sage:chi.jacobi_sum(n)
