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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(891, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([14,27]))
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pari:[g,chi] = znchar(Mod(670,891))
| Modulus: | \(891\) | |
| Conductor: | \(891\) |
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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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\(\chi_{891}(43,\cdot)\)
\(\chi_{891}(76,\cdot)\)
\(\chi_{891}(142,\cdot)\)
\(\chi_{891}(175,\cdot)\)
\(\chi_{891}(241,\cdot)\)
\(\chi_{891}(274,\cdot)\)
\(\chi_{891}(340,\cdot)\)
\(\chi_{891}(373,\cdot)\)
\(\chi_{891}(439,\cdot)\)
\(\chi_{891}(472,\cdot)\)
\(\chi_{891}(538,\cdot)\)
\(\chi_{891}(571,\cdot)\)
\(\chi_{891}(637,\cdot)\)
\(\chi_{891}(670,\cdot)\)
\(\chi_{891}(736,\cdot)\)
\(\chi_{891}(769,\cdot)\)
\(\chi_{891}(835,\cdot)\)
\(\chi_{891}(868,\cdot)\)
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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 ]
\((650,244)\) → \((e\left(\frac{7}{27}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 891 }(670, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{1}{18}\right)\) |
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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)
