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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(153, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([40,9]))
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pari:[g,chi] = znchar(Mod(95,153))
| Modulus: | \(153\) | |
| Conductor: | \(153\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(48\) |
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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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\(\chi_{153}(5,\cdot)\)
\(\chi_{153}(11,\cdot)\)
\(\chi_{153}(14,\cdot)\)
\(\chi_{153}(20,\cdot)\)
\(\chi_{153}(23,\cdot)\)
\(\chi_{153}(29,\cdot)\)
\(\chi_{153}(41,\cdot)\)
\(\chi_{153}(56,\cdot)\)
\(\chi_{153}(65,\cdot)\)
\(\chi_{153}(74,\cdot)\)
\(\chi_{153}(92,\cdot)\)
\(\chi_{153}(95,\cdot)\)
\(\chi_{153}(113,\cdot)\)
\(\chi_{153}(122,\cdot)\)
\(\chi_{153}(131,\cdot)\)
\(\chi_{153}(146,\cdot)\)
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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 ]
\((137,37)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{3}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 153 }(95, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{5}{6}\right)\) |
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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)
