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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(716, base_ring=CyclotomicField(178))
M = H._module
chi = DirichletCharacter(H, M([89,52]))
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pari:[g,chi] = znchar(Mod(139,716))
| Modulus: | \(716\) | |
| Conductor: | \(716\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(178\) |
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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)
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\(\chi_{716}(3,\cdot)\)
\(\chi_{716}(15,\cdot)\)
\(\chi_{716}(19,\cdot)\)
\(\chi_{716}(27,\cdot)\)
\(\chi_{716}(31,\cdot)\)
\(\chi_{716}(39,\cdot)\)
\(\chi_{716}(43,\cdot)\)
\(\chi_{716}(47,\cdot)\)
\(\chi_{716}(51,\cdot)\)
\(\chi_{716}(59,\cdot)\)
\(\chi_{716}(67,\cdot)\)
\(\chi_{716}(75,\cdot)\)
\(\chi_{716}(83,\cdot)\)
\(\chi_{716}(87,\cdot)\)
\(\chi_{716}(95,\cdot)\)
\(\chi_{716}(107,\cdot)\)
\(\chi_{716}(135,\cdot)\)
\(\chi_{716}(139,\cdot)\)
\(\chi_{716}(147,\cdot)\)
\(\chi_{716}(151,\cdot)\)
\(\chi_{716}(155,\cdot)\)
\(\chi_{716}(171,\cdot)\)
\(\chi_{716}(183,\cdot)\)
\(\chi_{716}(191,\cdot)\)
\(\chi_{716}(195,\cdot)\)
\(\chi_{716}(199,\cdot)\)
\(\chi_{716}(215,\cdot)\)
\(\chi_{716}(227,\cdot)\)
\(\chi_{716}(231,\cdot)\)
\(\chi_{716}(235,\cdot)\)
...
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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 ]
\((359,181)\) → \((-1,e\left(\frac{26}{89}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 716 }(139, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{9}{178}\right)\) | \(e\left(\frac{28}{89}\right)\) | \(e\left(\frac{81}{178}\right)\) | \(e\left(\frac{9}{89}\right)\) | \(e\left(\frac{157}{178}\right)\) | \(e\left(\frac{27}{89}\right)\) | \(e\left(\frac{65}{178}\right)\) | \(e\left(\frac{44}{89}\right)\) | \(e\left(\frac{49}{178}\right)\) | \(e\left(\frac{45}{89}\right)\) |
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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)
