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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(15730, base_ring=CyclotomicField(220))
M = H._module
chi = DirichletCharacter(H, M([55,212,165]))
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pari:[g,chi] = znchar(Mod(1747,15730))
\(\chi_{15730}(203,\cdot)\)
\(\chi_{15730}(317,\cdot)\)
\(\chi_{15730}(333,\cdot)\)
\(\chi_{15730}(577,\cdot)\)
\(\chi_{15730}(707,\cdot)\)
\(\chi_{15730}(983,\cdot)\)
\(\chi_{15730}(1357,\cdot)\)
\(\chi_{15730}(1373,\cdot)\)
\(\chi_{15730}(1633,\cdot)\)
\(\chi_{15730}(1747,\cdot)\)
\(\chi_{15730}(1763,\cdot)\)
\(\chi_{15730}(2007,\cdot)\)
\(\chi_{15730}(2137,\cdot)\)
\(\chi_{15730}(2413,\cdot)\)
\(\chi_{15730}(2787,\cdot)\)
\(\chi_{15730}(2803,\cdot)\)
\(\chi_{15730}(3063,\cdot)\)
\(\chi_{15730}(3177,\cdot)\)
\(\chi_{15730}(3193,\cdot)\)
\(\chi_{15730}(3437,\cdot)\)
\(\chi_{15730}(3567,\cdot)\)
\(\chi_{15730}(3843,\cdot)\)
\(\chi_{15730}(4217,\cdot)\)
\(\chi_{15730}(4233,\cdot)\)
\(\chi_{15730}(4493,\cdot)\)
\(\chi_{15730}(4623,\cdot)\)
\(\chi_{15730}(4997,\cdot)\)
\(\chi_{15730}(5273,\cdot)\)
\(\chi_{15730}(5663,\cdot)\)
\(\chi_{15730}(5923,\cdot)\)
...
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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 ]
\((3147,3511,1211)\) → \((i,e\left(\frac{53}{55}\right),-i)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 15730 }(1747, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{213}{220}\right)\) | \(e\left(\frac{51}{220}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{137}{220}\right)\) |
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sage:chi.jacobi_sum(n)
