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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2535, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([0,0,16]))
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pari:[g,chi] = znchar(Mod(451,2535))
\(\chi_{2535}(16,\cdot)\)
\(\chi_{2535}(61,\cdot)\)
\(\chi_{2535}(211,\cdot)\)
\(\chi_{2535}(256,\cdot)\)
\(\chi_{2535}(406,\cdot)\)
\(\chi_{2535}(451,\cdot)\)
\(\chi_{2535}(601,\cdot)\)
\(\chi_{2535}(646,\cdot)\)
\(\chi_{2535}(796,\cdot)\)
\(\chi_{2535}(841,\cdot)\)
\(\chi_{2535}(1186,\cdot)\)
\(\chi_{2535}(1231,\cdot)\)
\(\chi_{2535}(1381,\cdot)\)
\(\chi_{2535}(1426,\cdot)\)
\(\chi_{2535}(1576,\cdot)\)
\(\chi_{2535}(1621,\cdot)\)
\(\chi_{2535}(1771,\cdot)\)
\(\chi_{2535}(1816,\cdot)\)
\(\chi_{2535}(1966,\cdot)\)
\(\chi_{2535}(2011,\cdot)\)
\(\chi_{2535}(2161,\cdot)\)
\(\chi_{2535}(2206,\cdot)\)
\(\chi_{2535}(2356,\cdot)\)
\(\chi_{2535}(2401,\cdot)\)
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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 ]
\((1691,1522,1861)\) → \((1,1,e\left(\frac{8}{39}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
| \( \chi_{ 2535 }(451, a) \) |
\(1\) | \(1\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) |
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sage:chi.jacobi_sum(n)
