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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(736, base_ring=CyclotomicField(88))
M = H._module
chi = DirichletCharacter(H, M([0,33,64]))
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pari:[g,chi] = znchar(Mod(509,736))
| Modulus: | \(736\) | |
| Conductor: | \(736\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(88\) |
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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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{736}(13,\cdot)\)
\(\chi_{736}(29,\cdot)\)
\(\chi_{736}(77,\cdot)\)
\(\chi_{736}(85,\cdot)\)
\(\chi_{736}(101,\cdot)\)
\(\chi_{736}(117,\cdot)\)
\(\chi_{736}(133,\cdot)\)
\(\chi_{736}(141,\cdot)\)
\(\chi_{736}(165,\cdot)\)
\(\chi_{736}(173,\cdot)\)
\(\chi_{736}(197,\cdot)\)
\(\chi_{736}(213,\cdot)\)
\(\chi_{736}(261,\cdot)\)
\(\chi_{736}(269,\cdot)\)
\(\chi_{736}(285,\cdot)\)
\(\chi_{736}(301,\cdot)\)
\(\chi_{736}(317,\cdot)\)
\(\chi_{736}(325,\cdot)\)
\(\chi_{736}(349,\cdot)\)
\(\chi_{736}(357,\cdot)\)
\(\chi_{736}(381,\cdot)\)
\(\chi_{736}(397,\cdot)\)
\(\chi_{736}(445,\cdot)\)
\(\chi_{736}(453,\cdot)\)
\(\chi_{736}(469,\cdot)\)
\(\chi_{736}(485,\cdot)\)
\(\chi_{736}(501,\cdot)\)
\(\chi_{736}(509,\cdot)\)
\(\chi_{736}(533,\cdot)\)
\(\chi_{736}(541,\cdot)\)
...
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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 ]
\((415,645,97)\) → \((1,e\left(\frac{3}{8}\right),e\left(\frac{8}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 736 }(509, a) \) |
\(1\) | \(1\) | \(e\left(\frac{67}{88}\right)\) | \(e\left(\frac{9}{88}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{37}{88}\right)\) | \(e\left(\frac{71}{88}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{47}{88}\right)\) | \(e\left(\frac{29}{88}\right)\) |
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sage:chi.jacobi_sum(n)
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sage:chi.gauss_sum(a)
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pari:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
