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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([0,55,12]))
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pari:[g,chi] = znchar(Mod(309,736))
| 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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
|
\(\chi_{736}(5,\cdot)\)
\(\chi_{736}(21,\cdot)\)
\(\chi_{736}(37,\cdot)\)
\(\chi_{736}(53,\cdot)\)
\(\chi_{736}(61,\cdot)\)
\(\chi_{736}(109,\cdot)\)
\(\chi_{736}(125,\cdot)\)
\(\chi_{736}(149,\cdot)\)
\(\chi_{736}(157,\cdot)\)
\(\chi_{736}(181,\cdot)\)
\(\chi_{736}(189,\cdot)\)
\(\chi_{736}(205,\cdot)\)
\(\chi_{736}(221,\cdot)\)
\(\chi_{736}(237,\cdot)\)
\(\chi_{736}(245,\cdot)\)
\(\chi_{736}(293,\cdot)\)
\(\chi_{736}(309,\cdot)\)
\(\chi_{736}(333,\cdot)\)
\(\chi_{736}(341,\cdot)\)
\(\chi_{736}(365,\cdot)\)
\(\chi_{736}(373,\cdot)\)
\(\chi_{736}(389,\cdot)\)
\(\chi_{736}(405,\cdot)\)
\(\chi_{736}(421,\cdot)\)
\(\chi_{736}(429,\cdot)\)
\(\chi_{736}(477,\cdot)\)
\(\chi_{736}(493,\cdot)\)
\(\chi_{736}(517,\cdot)\)
\(\chi_{736}(525,\cdot)\)
\(\chi_{736}(549,\cdot)\)
...
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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 ]
\((415,645,97)\) → \((1,e\left(\frac{5}{8}\right),e\left(\frac{3}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 736 }(309, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{5}{88}\right)\) | \(e\left(\frac{67}{88}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{31}{88}\right)\) | \(e\left(\frac{25}{88}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{37}{88}\right)\) | \(e\left(\frac{79}{88}\right)\) |
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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)
