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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2864, base_ring=CyclotomicField(356))
M = H._module
chi = DirichletCharacter(H, M([0,267,342]))
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pari:[g,chi] = znchar(Mod(365,2864))
| Modulus: | \(2864\) | |
| Conductor: | \(2864\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(356\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
|
| Real: | no |
| Primitive: | yes |
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sage:chi.is_primitive()
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pari:#znconreyconductor(g,chi)==1
|
| Minimal: | yes |
| Parity: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
|
\(\chi_{2864}(21,\cdot)\)
\(\chi_{2864}(37,\cdot)\)
\(\chi_{2864}(53,\cdot)\)
\(\chi_{2864}(69,\cdot)\)
\(\chi_{2864}(109,\cdot)\)
\(\chi_{2864}(133,\cdot)\)
\(\chi_{2864}(157,\cdot)\)
\(\chi_{2864}(165,\cdot)\)
\(\chi_{2864}(181,\cdot)\)
\(\chi_{2864}(189,\cdot)\)
\(\chi_{2864}(197,\cdot)\)
\(\chi_{2864}(205,\cdot)\)
\(\chi_{2864}(213,\cdot)\)
\(\chi_{2864}(229,\cdot)\)
\(\chi_{2864}(237,\cdot)\)
\(\chi_{2864}(269,\cdot)\)
\(\chi_{2864}(277,\cdot)\)
\(\chi_{2864}(293,\cdot)\)
\(\chi_{2864}(301,\cdot)\)
\(\chi_{2864}(309,\cdot)\)
\(\chi_{2864}(333,\cdot)\)
\(\chi_{2864}(341,\cdot)\)
\(\chi_{2864}(349,\cdot)\)
\(\chi_{2864}(365,\cdot)\)
\(\chi_{2864}(381,\cdot)\)
\(\chi_{2864}(413,\cdot)\)
\(\chi_{2864}(421,\cdot)\)
\(\chi_{2864}(429,\cdot)\)
\(\chi_{2864}(437,\cdot)\)
\(\chi_{2864}(461,\cdot)\)
...
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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 ]
\((1791,2149,897)\) → \((1,-i,e\left(\frac{171}{178}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 2864 }(365, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{356}\right)\) | \(e\left(\frac{115}{356}\right)\) | \(e\left(\frac{69}{89}\right)\) | \(e\left(\frac{1}{178}\right)\) | \(e\left(\frac{57}{356}\right)\) | \(e\left(\frac{273}{356}\right)\) | \(e\left(\frac{29}{89}\right)\) | \(e\left(\frac{42}{89}\right)\) | \(e\left(\frac{45}{356}\right)\) | \(e\left(\frac{277}{356}\right)\) |
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sage:chi.jacobi_sum(n)
