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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([0,55,0,47]))
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pari:[g,chi] = znchar(Mod(821,7260))
\(\chi_{7260}(41,\cdot)\)
\(\chi_{7260}(101,\cdot)\)
\(\chi_{7260}(281,\cdot)\)
\(\chi_{7260}(701,\cdot)\)
\(\chi_{7260}(761,\cdot)\)
\(\chi_{7260}(821,\cdot)\)
\(\chi_{7260}(1361,\cdot)\)
\(\chi_{7260}(1421,\cdot)\)
\(\chi_{7260}(1481,\cdot)\)
\(\chi_{7260}(1601,\cdot)\)
\(\chi_{7260}(2021,\cdot)\)
\(\chi_{7260}(2081,\cdot)\)
\(\chi_{7260}(2141,\cdot)\)
\(\chi_{7260}(2261,\cdot)\)
\(\chi_{7260}(2681,\cdot)\)
\(\chi_{7260}(2741,\cdot)\)
\(\chi_{7260}(2801,\cdot)\)
\(\chi_{7260}(2921,\cdot)\)
\(\chi_{7260}(3341,\cdot)\)
\(\chi_{7260}(3401,\cdot)\)
\(\chi_{7260}(3461,\cdot)\)
\(\chi_{7260}(3581,\cdot)\)
\(\chi_{7260}(4001,\cdot)\)
\(\chi_{7260}(4061,\cdot)\)
\(\chi_{7260}(4121,\cdot)\)
\(\chi_{7260}(4241,\cdot)\)
\(\chi_{7260}(4661,\cdot)\)
\(\chi_{7260}(4721,\cdot)\)
\(\chi_{7260}(4781,\cdot)\)
\(\chi_{7260}(4901,\cdot)\)
...
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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 ]
\((3631,4841,4357,7141)\) → \((1,-1,1,e\left(\frac{47}{110}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 7260 }(821, a) \) |
\(1\) | \(1\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{51}{110}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{15}{22}\right)\) |
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sage:chi.jacobi_sum(n)
