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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(31360, base_ring=CyclotomicField(336))
M = H._module
chi = DirichletCharacter(H, M([0,105,84,88]))
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pari:[g,chi] = znchar(Mod(6137,31360))
\(\chi_{31360}(73,\cdot)\)
\(\chi_{31360}(537,\cdot)\)
\(\chi_{31360}(857,\cdot)\)
\(\chi_{31360}(873,\cdot)\)
\(\chi_{31360}(1193,\cdot)\)
\(\chi_{31360}(1657,\cdot)\)
\(\chi_{31360}(1977,\cdot)\)
\(\chi_{31360}(1993,\cdot)\)
\(\chi_{31360}(2313,\cdot)\)
\(\chi_{31360}(2777,\cdot)\)
\(\chi_{31360}(3097,\cdot)\)
\(\chi_{31360}(3113,\cdot)\)
\(\chi_{31360}(3433,\cdot)\)
\(\chi_{31360}(3897,\cdot)\)
\(\chi_{31360}(4217,\cdot)\)
\(\chi_{31360}(4553,\cdot)\)
\(\chi_{31360}(5337,\cdot)\)
\(\chi_{31360}(5353,\cdot)\)
\(\chi_{31360}(5673,\cdot)\)
\(\chi_{31360}(6137,\cdot)\)
\(\chi_{31360}(6457,\cdot)\)
\(\chi_{31360}(6473,\cdot)\)
\(\chi_{31360}(7257,\cdot)\)
\(\chi_{31360}(7593,\cdot)\)
\(\chi_{31360}(7913,\cdot)\)
\(\chi_{31360}(8377,\cdot)\)
\(\chi_{31360}(8697,\cdot)\)
\(\chi_{31360}(8713,\cdot)\)
\(\chi_{31360}(9033,\cdot)\)
\(\chi_{31360}(9497,\cdot)\)
...
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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 ]
\((17151,28421,18817,10881)\) → \((1,e\left(\frac{5}{16}\right),i,e\left(\frac{11}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 31360 }(6137, a) \) |
\(1\) | \(1\) | \(e\left(\frac{319}{336}\right)\) | \(e\left(\frac{151}{168}\right)\) | \(e\left(\frac{13}{336}\right)\) | \(e\left(\frac{9}{112}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{13}{168}\right)\) | \(e\left(\frac{95}{112}\right)\) | \(e\left(\frac{73}{112}\right)\) | \(e\left(\frac{1}{3}\right)\) |
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sage:chi.jacobi_sum(n)
