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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(845, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([117,41]))
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pari:[g,chi] = znchar(Mod(58,845))
| Modulus: | \(845\) | |
| Conductor: | \(845\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(156\) |
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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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\(\chi_{845}(7,\cdot)\)
\(\chi_{845}(28,\cdot)\)
\(\chi_{845}(37,\cdot)\)
\(\chi_{845}(58,\cdot)\)
\(\chi_{845}(72,\cdot)\)
\(\chi_{845}(93,\cdot)\)
\(\chi_{845}(102,\cdot)\)
\(\chi_{845}(123,\cdot)\)
\(\chi_{845}(137,\cdot)\)
\(\chi_{845}(158,\cdot)\)
\(\chi_{845}(167,\cdot)\)
\(\chi_{845}(202,\cdot)\)
\(\chi_{845}(223,\cdot)\)
\(\chi_{845}(232,\cdot)\)
\(\chi_{845}(253,\cdot)\)
\(\chi_{845}(267,\cdot)\)
\(\chi_{845}(288,\cdot)\)
\(\chi_{845}(297,\cdot)\)
\(\chi_{845}(318,\cdot)\)
\(\chi_{845}(332,\cdot)\)
\(\chi_{845}(353,\cdot)\)
\(\chi_{845}(362,\cdot)\)
\(\chi_{845}(383,\cdot)\)
\(\chi_{845}(397,\cdot)\)
\(\chi_{845}(448,\cdot)\)
\(\chi_{845}(462,\cdot)\)
\(\chi_{845}(483,\cdot)\)
\(\chi_{845}(492,\cdot)\)
\(\chi_{845}(513,\cdot)\)
\(\chi_{845}(527,\cdot)\)
...
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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 ]
\((677,171)\) → \((-i,e\left(\frac{41}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 845 }(58, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{131}{156}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{133}{156}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{11}{156}\right)\) | \(e\left(\frac{45}{52}\right)\) | \(e\left(\frac{23}{26}\right)\) |
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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)
