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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(97020, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,35,0,5,21]))
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pari:[g,chi] = znchar(Mod(69431,97020))
\(\chi_{97020}(131,\cdot)\)
\(\chi_{97020}(13331,\cdot)\)
\(\chi_{97020}(13991,\cdot)\)
\(\chi_{97020}(27191,\cdot)\)
\(\chi_{97020}(41051,\cdot)\)
\(\chi_{97020}(41711,\cdot)\)
\(\chi_{97020}(55571,\cdot)\)
\(\chi_{97020}(68771,\cdot)\)
\(\chi_{97020}(69431,\cdot)\)
\(\chi_{97020}(82631,\cdot)\)
\(\chi_{97020}(83291,\cdot)\)
\(\chi_{97020}(96491,\cdot)\)
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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 ]
\((48511,43121,77617,9901,44101)\) → \((-1,e\left(\frac{5}{6}\right),1,e\left(\frac{5}{42}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 97020 }(69431, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(1\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{13}{14}\right)\) |
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sage:chi.jacobi_sum(n)
