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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(511, base_ring=CyclotomicField(72))
M = H._module
chi = DirichletCharacter(H, M([36,41]))
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pari:[g,chi] = znchar(Mod(160,511))
| Modulus: | \(511\) | |
| Conductor: | \(511\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(72\) |
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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
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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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\(\chi_{511}(13,\cdot)\)
\(\chi_{511}(20,\cdot)\)
\(\chi_{511}(34,\cdot)\)
\(\chi_{511}(62,\cdot)\)
\(\chi_{511}(104,\cdot)\)
\(\chi_{511}(118,\cdot)\)
\(\chi_{511}(132,\cdot)\)
\(\chi_{511}(160,\cdot)\)
\(\chi_{511}(174,\cdot)\)
\(\chi_{511}(188,\cdot)\)
\(\chi_{511}(230,\cdot)\)
\(\chi_{511}(258,\cdot)\)
\(\chi_{511}(272,\cdot)\)
\(\chi_{511}(279,\cdot)\)
\(\chi_{511}(307,\cdot)\)
\(\chi_{511}(321,\cdot)\)
\(\chi_{511}(370,\cdot)\)
\(\chi_{511}(391,\cdot)\)
\(\chi_{511}(398,\cdot)\)
\(\chi_{511}(405,\cdot)\)
\(\chi_{511}(412,\cdot)\)
\(\chi_{511}(433,\cdot)\)
\(\chi_{511}(482,\cdot)\)
\(\chi_{511}(496,\cdot)\)
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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 ]
\((220,78)\) → \((-1,e\left(\frac{41}{72}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 511 }(160, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{5}{72}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{23}{72}\right)\) | \(e\left(\frac{1}{36}\right)\) |
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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)
