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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(31360, base_ring=CyclotomicField(112))
M = H._module
chi = DirichletCharacter(H, M([56,21,0,96]))
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pari:[g,chi] = znchar(Mod(14071,31360))
\(\chi_{31360}(71,\cdot)\)
\(\chi_{31360}(631,\cdot)\)
\(\chi_{31360}(1191,\cdot)\)
\(\chi_{31360}(1751,\cdot)\)
\(\chi_{31360}(2311,\cdot)\)
\(\chi_{31360}(2871,\cdot)\)
\(\chi_{31360}(3991,\cdot)\)
\(\chi_{31360}(4551,\cdot)\)
\(\chi_{31360}(5111,\cdot)\)
\(\chi_{31360}(5671,\cdot)\)
\(\chi_{31360}(6231,\cdot)\)
\(\chi_{31360}(6791,\cdot)\)
\(\chi_{31360}(7911,\cdot)\)
\(\chi_{31360}(8471,\cdot)\)
\(\chi_{31360}(9031,\cdot)\)
\(\chi_{31360}(9591,\cdot)\)
\(\chi_{31360}(10151,\cdot)\)
\(\chi_{31360}(10711,\cdot)\)
\(\chi_{31360}(11831,\cdot)\)
\(\chi_{31360}(12391,\cdot)\)
\(\chi_{31360}(12951,\cdot)\)
\(\chi_{31360}(13511,\cdot)\)
\(\chi_{31360}(14071,\cdot)\)
\(\chi_{31360}(14631,\cdot)\)
\(\chi_{31360}(15751,\cdot)\)
\(\chi_{31360}(16311,\cdot)\)
\(\chi_{31360}(16871,\cdot)\)
\(\chi_{31360}(17431,\cdot)\)
\(\chi_{31360}(17991,\cdot)\)
\(\chi_{31360}(18551,\cdot)\)
...
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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 ]
\((17151,28421,18817,10881)\) → \((-1,e\left(\frac{3}{16}\right),1,e\left(\frac{6}{7}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 31360 }(14071, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{103}{112}\right)\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{81}{112}\right)\) | \(e\left(\frac{11}{112}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{85}{112}\right)\) | \(e\left(\frac{55}{112}\right)\) | \(1\) |
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sage:chi.jacobi_sum(n)
