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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(34272, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,0,0,16,15]))
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pari:[g,chi] = znchar(Mod(20575,34272))
\(\chi_{34272}(415,\cdot)\)
\(\chi_{34272}(991,\cdot)\)
\(\chi_{34272}(4447,\cdot)\)
\(\chi_{34272}(6463,\cdot)\)
\(\chi_{34272}(9055,\cdot)\)
\(\chi_{34272}(10495,\cdot)\)
\(\chi_{34272}(13087,\cdot)\)
\(\chi_{34272}(15103,\cdot)\)
\(\chi_{34272}(18559,\cdot)\)
\(\chi_{34272}(19135,\cdot)\)
\(\chi_{34272}(20575,\cdot)\)
\(\chi_{34272}(21151,\cdot)\)
\(\chi_{34272}(25183,\cdot)\)
\(\chi_{34272}(28639,\cdot)\)
\(\chi_{34272}(32671,\cdot)\)
\(\chi_{34272}(33247,\cdot)\)
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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 ]
\((2143,29989,3809,14689,14113)\) → \((-1,1,1,e\left(\frac{1}{3}\right),e\left(\frac{5}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 34272 }(20575, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(i\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{7}{16}\right)\) |
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sage:chi.jacobi_sum(n)
