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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(608, base_ring=CyclotomicField(72))
M = H._module
chi = DirichletCharacter(H, M([0,27,8]))
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pari:[g,chi] = znchar(Mod(61,608))
| Modulus: | \(608\) | |
| Conductor: | \(608\) |
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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
|
| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{608}(5,\cdot)\)
\(\chi_{608}(61,\cdot)\)
\(\chi_{608}(85,\cdot)\)
\(\chi_{608}(93,\cdot)\)
\(\chi_{608}(101,\cdot)\)
\(\chi_{608}(149,\cdot)\)
\(\chi_{608}(157,\cdot)\)
\(\chi_{608}(213,\cdot)\)
\(\chi_{608}(237,\cdot)\)
\(\chi_{608}(245,\cdot)\)
\(\chi_{608}(253,\cdot)\)
\(\chi_{608}(301,\cdot)\)
\(\chi_{608}(309,\cdot)\)
\(\chi_{608}(365,\cdot)\)
\(\chi_{608}(389,\cdot)\)
\(\chi_{608}(397,\cdot)\)
\(\chi_{608}(405,\cdot)\)
\(\chi_{608}(453,\cdot)\)
\(\chi_{608}(461,\cdot)\)
\(\chi_{608}(517,\cdot)\)
\(\chi_{608}(541,\cdot)\)
\(\chi_{608}(549,\cdot)\)
\(\chi_{608}(557,\cdot)\)
\(\chi_{608}(605,\cdot)\)
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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 ]
\((191,229,97)\) → \((1,e\left(\frac{3}{8}\right),e\left(\frac{1}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
| \( \chi_{ 608 }(61, a) \) |
\(1\) | \(1\) | \(e\left(\frac{41}{72}\right)\) | \(e\left(\frac{11}{72}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{13}{72}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{71}{72}\right)\) | \(e\left(\frac{17}{36}\right)\) |
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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)
