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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(132300, base_ring=CyclotomicField(420))
M = H._module
chi = DirichletCharacter(H, M([0,280,189,400]))
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pari:[g,chi] = znchar(Mod(3637,132300))
\(\chi_{132300}(37,\cdot)\)
\(\chi_{132300}(613,\cdot)\)
\(\chi_{132300}(3637,\cdot)\)
\(\chi_{132300}(3817,\cdot)\)
\(\chi_{132300}(4573,\cdot)\)
\(\chi_{132300}(7597,\cdot)\)
\(\chi_{132300}(8173,\cdot)\)
\(\chi_{132300}(8353,\cdot)\)
\(\chi_{132300}(11197,\cdot)\)
\(\chi_{132300}(11377,\cdot)\)
\(\chi_{132300}(11953,\cdot)\)
\(\chi_{132300}(14977,\cdot)\)
\(\chi_{132300}(15733,\cdot)\)
\(\chi_{132300}(15913,\cdot)\)
\(\chi_{132300}(18937,\cdot)\)
\(\chi_{132300}(19513,\cdot)\)
\(\chi_{132300}(22537,\cdot)\)
\(\chi_{132300}(23473,\cdot)\)
\(\chi_{132300}(26317,\cdot)\)
\(\chi_{132300}(26497,\cdot)\)
\(\chi_{132300}(27073,\cdot)\)
\(\chi_{132300}(27253,\cdot)\)
\(\chi_{132300}(30097,\cdot)\)
\(\chi_{132300}(30277,\cdot)\)
\(\chi_{132300}(30853,\cdot)\)
\(\chi_{132300}(31033,\cdot)\)
\(\chi_{132300}(34633,\cdot)\)
\(\chi_{132300}(34813,\cdot)\)
\(\chi_{132300}(37837,\cdot)\)
\(\chi_{132300}(38413,\cdot)\)
...
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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 ]
\((66151,122501,15877,54001)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{9}{20}\right),e\left(\frac{20}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 132300 }(3637, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{101}{105}\right)\) | \(e\left(\frac{131}{420}\right)\) | \(e\left(\frac{277}{420}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{199}{420}\right)\) | \(e\left(\frac{149}{210}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{221}{420}\right)\) | \(e\left(\frac{44}{105}\right)\) | \(e\left(\frac{11}{84}\right)\) |
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sage:chi.jacobi_sum(n)
