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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(25410, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([0,165,275,177]))
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pari:[g,chi] = znchar(Mod(4219,25410))
\(\chi_{25410}(19,\cdot)\)
\(\chi_{25410}(409,\cdot)\)
\(\chi_{25410}(1069,\cdot)\)
\(\chi_{25410}(1249,\cdot)\)
\(\chi_{25410}(1459,\cdot)\)
\(\chi_{25410}(1669,\cdot)\)
\(\chi_{25410}(2119,\cdot)\)
\(\chi_{25410}(2329,\cdot)\)
\(\chi_{25410}(2719,\cdot)\)
\(\chi_{25410}(3559,\cdot)\)
\(\chi_{25410}(3769,\cdot)\)
\(\chi_{25410}(3979,\cdot)\)
\(\chi_{25410}(4219,\cdot)\)
\(\chi_{25410}(4429,\cdot)\)
\(\chi_{25410}(4639,\cdot)\)
\(\chi_{25410}(5029,\cdot)\)
\(\chi_{25410}(5689,\cdot)\)
\(\chi_{25410}(5869,\cdot)\)
\(\chi_{25410}(6079,\cdot)\)
\(\chi_{25410}(6529,\cdot)\)
\(\chi_{25410}(6739,\cdot)\)
\(\chi_{25410}(6949,\cdot)\)
\(\chi_{25410}(7339,\cdot)\)
\(\chi_{25410}(7999,\cdot)\)
\(\chi_{25410}(8179,\cdot)\)
\(\chi_{25410}(8599,\cdot)\)
\(\chi_{25410}(8839,\cdot)\)
\(\chi_{25410}(9049,\cdot)\)
\(\chi_{25410}(9259,\cdot)\)
\(\chi_{25410}(9649,\cdot)\)
...
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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 ]
\((8471,15247,14521,7141)\) → \((1,-1,e\left(\frac{5}{6}\right),e\left(\frac{59}{110}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 25410 }(4219, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{110}\right)\) | \(e\left(\frac{203}{330}\right)\) | \(e\left(\frac{113}{165}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{317}{330}\right)\) | \(e\left(\frac{229}{330}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{38}{165}\right)\) |
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sage:chi.jacobi_sum(n)
