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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(920, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,22,11,36]))
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pari:[g,chi] = znchar(Mod(627,920))
| Modulus: | \(920\) | |
| Conductor: | \(920\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(44\) |
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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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\(\chi_{920}(3,\cdot)\)
\(\chi_{920}(27,\cdot)\)
\(\chi_{920}(123,\cdot)\)
\(\chi_{920}(147,\cdot)\)
\(\chi_{920}(163,\cdot)\)
\(\chi_{920}(187,\cdot)\)
\(\chi_{920}(243,\cdot)\)
\(\chi_{920}(307,\cdot)\)
\(\chi_{920}(347,\cdot)\)
\(\chi_{920}(363,\cdot)\)
\(\chi_{920}(403,\cdot)\)
\(\chi_{920}(427,\cdot)\)
\(\chi_{920}(443,\cdot)\)
\(\chi_{920}(547,\cdot)\)
\(\chi_{920}(587,\cdot)\)
\(\chi_{920}(627,\cdot)\)
\(\chi_{920}(683,\cdot)\)
\(\chi_{920}(763,\cdot)\)
\(\chi_{920}(867,\cdot)\)
\(\chi_{920}(883,\cdot)\)
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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 ]
\((231,461,737,281)\) → \((-1,-1,i,e\left(\frac{9}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 920 }(627, a) \) |
\(1\) | \(1\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{8}{11}\right)\) |
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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)
