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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7260, base_ring=CyclotomicField(220))
M = H._module
chi = DirichletCharacter(H, M([0,110,165,212]))
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pari:[g,chi] = znchar(Mod(53,7260))
\(\chi_{7260}(53,\cdot)\)
\(\chi_{7260}(113,\cdot)\)
\(\chi_{7260}(137,\cdot)\)
\(\chi_{7260}(257,\cdot)\)
\(\chi_{7260}(317,\cdot)\)
\(\chi_{7260}(377,\cdot)\)
\(\chi_{7260}(533,\cdot)\)
\(\chi_{7260}(653,\cdot)\)
\(\chi_{7260}(713,\cdot)\)
\(\chi_{7260}(773,\cdot)\)
\(\chi_{7260}(797,\cdot)\)
\(\chi_{7260}(917,\cdot)\)
\(\chi_{7260}(1037,\cdot)\)
\(\chi_{7260}(1193,\cdot)\)
\(\chi_{7260}(1313,\cdot)\)
\(\chi_{7260}(1373,\cdot)\)
\(\chi_{7260}(1433,\cdot)\)
\(\chi_{7260}(1457,\cdot)\)
\(\chi_{7260}(1577,\cdot)\)
\(\chi_{7260}(1637,\cdot)\)
\(\chi_{7260}(1853,\cdot)\)
\(\chi_{7260}(1973,\cdot)\)
\(\chi_{7260}(2033,\cdot)\)
\(\chi_{7260}(2093,\cdot)\)
\(\chi_{7260}(2117,\cdot)\)
\(\chi_{7260}(2237,\cdot)\)
\(\chi_{7260}(2297,\cdot)\)
\(\chi_{7260}(2357,\cdot)\)
\(\chi_{7260}(2513,\cdot)\)
\(\chi_{7260}(2633,\cdot)\)
...
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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 ]
\((3631,4841,4357,7141)\) → \((1,-1,-i,e\left(\frac{53}{55}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 7260 }(53, a) \) |
\(1\) | \(1\) | \(e\left(\frac{109}{220}\right)\) | \(e\left(\frac{127}{220}\right)\) | \(e\left(\frac{103}{220}\right)\) | \(e\left(\frac{53}{110}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{49}{220}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{15}{44}\right)\) |
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sage:chi.jacobi_sum(n)
