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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(81, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([19]))
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pari:[g,chi] = znchar(Mod(56,81))
| Modulus: | \(81\) | |
| Conductor: | \(81\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(54\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{81}(2,\cdot)\)
\(\chi_{81}(5,\cdot)\)
\(\chi_{81}(11,\cdot)\)
\(\chi_{81}(14,\cdot)\)
\(\chi_{81}(20,\cdot)\)
\(\chi_{81}(23,\cdot)\)
\(\chi_{81}(29,\cdot)\)
\(\chi_{81}(32,\cdot)\)
\(\chi_{81}(38,\cdot)\)
\(\chi_{81}(41,\cdot)\)
\(\chi_{81}(47,\cdot)\)
\(\chi_{81}(50,\cdot)\)
\(\chi_{81}(56,\cdot)\)
\(\chi_{81}(59,\cdot)\)
\(\chi_{81}(65,\cdot)\)
\(\chi_{81}(68,\cdot)\)
\(\chi_{81}(74,\cdot)\)
\(\chi_{81}(77,\cdot)\)
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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 ]
\(2\) → \(e\left(\frac{19}{54}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 81 }(56, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{11}{27}\right)\) |
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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)
