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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(656, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,10,3]))
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pari:[g,chi] = znchar(Mod(11,656))
| Modulus: | \(656\) | |
| Conductor: | \(656\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(40\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
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| Real: | no |
| Primitive: | yes |
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sage:chi.is_primitive()
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pari:#znconreyconductor(g,chi)==1
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| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{656}(11,\cdot)\)
\(\chi_{656}(19,\cdot)\)
\(\chi_{656}(35,\cdot)\)
\(\chi_{656}(67,\cdot)\)
\(\chi_{656}(75,\cdot)\)
\(\chi_{656}(99,\cdot)\)
\(\chi_{656}(147,\cdot)\)
\(\chi_{656}(171,\cdot)\)
\(\chi_{656}(179,\cdot)\)
\(\chi_{656}(211,\cdot)\)
\(\chi_{656}(227,\cdot)\)
\(\chi_{656}(235,\cdot)\)
\(\chi_{656}(299,\cdot)\)
\(\chi_{656}(315,\cdot)\)
\(\chi_{656}(587,\cdot)\)
\(\chi_{656}(603,\cdot)\)
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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 ]
\((575,165,129)\) → \((-1,i,e\left(\frac{3}{40}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 656 }(11, a) \) |
\(1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(-i\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) |
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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)
