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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8216, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([78,0,13,126]))
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pari:[g,chi] = znchar(Mod(15,8216))
\(\chi_{8216}(15,\cdot)\)
\(\chi_{8216}(71,\cdot)\)
\(\chi_{8216}(175,\cdot)\)
\(\chi_{8216}(215,\cdot)\)
\(\chi_{8216}(535,\cdot)\)
\(\chi_{8216}(847,\cdot)\)
\(\chi_{8216}(1463,\cdot)\)
\(\chi_{8216}(1671,\cdot)\)
\(\chi_{8216}(1831,\cdot)\)
\(\chi_{8216}(2087,\cdot)\)
\(\chi_{8216}(2095,\cdot)\)
\(\chi_{8216}(2191,\cdot)\)
\(\chi_{8216}(2303,\cdot)\)
\(\chi_{8216}(2463,\cdot)\)
\(\chi_{8216}(2719,\cdot)\)
\(\chi_{8216}(2823,\cdot)\)
\(\chi_{8216}(2871,\cdot)\)
\(\chi_{8216}(3231,\cdot)\)
\(\chi_{8216}(3335,\cdot)\)
\(\chi_{8216}(3503,\cdot)\)
\(\chi_{8216}(3703,\cdot)\)
\(\chi_{8216}(3807,\cdot)\)
\(\chi_{8216}(3863,\cdot)\)
\(\chi_{8216}(3967,\cdot)\)
\(\chi_{8216}(4327,\cdot)\)
\(\chi_{8216}(4335,\cdot)\)
\(\chi_{8216}(4439,\cdot)\)
\(\chi_{8216}(4639,\cdot)\)
\(\chi_{8216}(4959,\cdot)\)
\(\chi_{8216}(5271,\cdot)\)
...
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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 ]
\((2055,4109,3161,3953)\) → \((-1,1,e\left(\frac{1}{12}\right),e\left(\frac{21}{26}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 8216 }(15, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{43}{52}\right)\) | \(e\left(\frac{35}{156}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{1}{156}\right)\) | \(e\left(\frac{73}{156}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{119}{156}\right)\) | \(e\left(\frac{45}{52}\right)\) | \(e\left(\frac{1}{3}\right)\) |
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sage:chi.jacobi_sum(n)
