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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(667, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([26,33]))
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pari:[g,chi] = znchar(Mod(481,667))
| Modulus: | \(667\) | |
| Conductor: | \(667\) |
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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)
|
| 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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\(\chi_{667}(17,\cdot)\)
\(\chi_{667}(99,\cdot)\)
\(\chi_{667}(157,\cdot)\)
\(\chi_{667}(191,\cdot)\)
\(\chi_{667}(244,\cdot)\)
\(\chi_{667}(249,\cdot)\)
\(\chi_{667}(273,\cdot)\)
\(\chi_{667}(336,\cdot)\)
\(\chi_{667}(360,\cdot)\)
\(\chi_{667}(365,\cdot)\)
\(\chi_{667}(389,\cdot)\)
\(\chi_{667}(447,\cdot)\)
\(\chi_{667}(452,\cdot)\)
\(\chi_{667}(481,\cdot)\)
\(\chi_{667}(534,\cdot)\)
\(\chi_{667}(539,\cdot)\)
\(\chi_{667}(563,\cdot)\)
\(\chi_{667}(592,\cdot)\)
\(\chi_{667}(626,\cdot)\)
\(\chi_{667}(655,\cdot)\)
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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 ]
\((465,553)\) → \((e\left(\frac{13}{22}\right),-i)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 667 }(481, a) \) |
\(1\) | \(1\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{3}{44}\right)\) |
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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)
