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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7260, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([55,0,0,51]))
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pari:[g,chi] = znchar(Mod(2851,7260))
\(\chi_{7260}(151,\cdot)\)
\(\chi_{7260}(211,\cdot)\)
\(\chi_{7260}(271,\cdot)\)
\(\chi_{7260}(391,\cdot)\)
\(\chi_{7260}(811,\cdot)\)
\(\chi_{7260}(871,\cdot)\)
\(\chi_{7260}(931,\cdot)\)
\(\chi_{7260}(1051,\cdot)\)
\(\chi_{7260}(1471,\cdot)\)
\(\chi_{7260}(1531,\cdot)\)
\(\chi_{7260}(1591,\cdot)\)
\(\chi_{7260}(1711,\cdot)\)
\(\chi_{7260}(2131,\cdot)\)
\(\chi_{7260}(2191,\cdot)\)
\(\chi_{7260}(2251,\cdot)\)
\(\chi_{7260}(2371,\cdot)\)
\(\chi_{7260}(2791,\cdot)\)
\(\chi_{7260}(2851,\cdot)\)
\(\chi_{7260}(2911,\cdot)\)
\(\chi_{7260}(3031,\cdot)\)
\(\chi_{7260}(3451,\cdot)\)
\(\chi_{7260}(3511,\cdot)\)
\(\chi_{7260}(3571,\cdot)\)
\(\chi_{7260}(3691,\cdot)\)
\(\chi_{7260}(4171,\cdot)\)
\(\chi_{7260}(4231,\cdot)\)
\(\chi_{7260}(4351,\cdot)\)
\(\chi_{7260}(4771,\cdot)\)
\(\chi_{7260}(4891,\cdot)\)
\(\chi_{7260}(5011,\cdot)\)
...
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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 ]
\((3631,4841,4357,7141)\) → \((-1,1,1,e\left(\frac{51}{110}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 7260 }(2851, a) \) |
\(1\) | \(1\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{1}{11}\right)\) |
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sage:chi.jacobi_sum(n)
