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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(845, base_ring=CyclotomicField(52))
M = H._module
chi = DirichletCharacter(H, M([39,22]))
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pari:[g,chi] = znchar(Mod(38,845))
| Modulus: | \(845\) | |
| Conductor: | \(845\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(52\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{845}(12,\cdot)\)
\(\chi_{845}(38,\cdot)\)
\(\chi_{845}(77,\cdot)\)
\(\chi_{845}(103,\cdot)\)
\(\chi_{845}(142,\cdot)\)
\(\chi_{845}(207,\cdot)\)
\(\chi_{845}(233,\cdot)\)
\(\chi_{845}(272,\cdot)\)
\(\chi_{845}(298,\cdot)\)
\(\chi_{845}(363,\cdot)\)
\(\chi_{845}(402,\cdot)\)
\(\chi_{845}(428,\cdot)\)
\(\chi_{845}(467,\cdot)\)
\(\chi_{845}(493,\cdot)\)
\(\chi_{845}(532,\cdot)\)
\(\chi_{845}(558,\cdot)\)
\(\chi_{845}(597,\cdot)\)
\(\chi_{845}(623,\cdot)\)
\(\chi_{845}(662,\cdot)\)
\(\chi_{845}(688,\cdot)\)
\(\chi_{845}(727,\cdot)\)
\(\chi_{845}(753,\cdot)\)
\(\chi_{845}(792,\cdot)\)
\(\chi_{845}(818,\cdot)\)
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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 ]
\((677,171)\) → \((-i,e\left(\frac{11}{26}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 845 }(38, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{9}{52}\right)\) | \(e\left(\frac{37}{52}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{1}{52}\right)\) | \(e\left(\frac{27}{52}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{3}{52}\right)\) | \(e\left(\frac{5}{26}\right)\) |
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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)
