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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(24648, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([0,0,78,13,150]))
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pari:[g,chi] = znchar(Mod(41,24648))
\(\chi_{24648}(41,\cdot)\)
\(\chi_{24648}(137,\cdot)\)
\(\chi_{24648}(665,\cdot)\)
\(\chi_{24648}(1649,\cdot)\)
\(\chi_{24648}(1913,\cdot)\)
\(\chi_{24648}(2273,\cdot)\)
\(\chi_{24648}(2585,\cdot)\)
\(\chi_{24648}(3569,\cdot)\)
\(\chi_{24648}(3833,\cdot)\)
\(\chi_{24648}(4457,\cdot)\)
\(\chi_{24648}(4673,\cdot)\)
\(\chi_{24648}(5441,\cdot)\)
\(\chi_{24648}(5705,\cdot)\)
\(\chi_{24648}(6065,\cdot)\)
\(\chi_{24648}(6233,\cdot)\)
\(\chi_{24648}(6377,\cdot)\)
\(\chi_{24648}(7625,\cdot)\)
\(\chi_{24648}(8249,\cdot)\)
\(\chi_{24648}(8465,\cdot)\)
\(\chi_{24648}(9497,\cdot)\)
\(\chi_{24648}(9665,\cdot)\)
\(\chi_{24648}(10025,\cdot)\)
\(\chi_{24648}(10601,\cdot)\)
\(\chi_{24648}(12257,\cdot)\)
\(\chi_{24648}(13409,\cdot)\)
\(\chi_{24648}(13457,\cdot)\)
\(\chi_{24648}(13817,\cdot)\)
\(\chi_{24648}(14393,\cdot)\)
\(\chi_{24648}(16049,\cdot)\)
\(\chi_{24648}(16841,\cdot)\)
...
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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 ]
\((18487,12325,16433,11377,12169)\) → \((1,1,-1,e\left(\frac{1}{12}\right),e\left(\frac{25}{26}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 24648 }(41, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{45}{52}\right)\) | \(e\left(\frac{137}{156}\right)\) | \(e\left(\frac{73}{156}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{29}{156}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{31}{52}\right)\) | \(e\left(\frac{29}{39}\right)\) |
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sage:chi.jacobi_sum(n)
