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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(845, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([39,28]))
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pari:[g,chi] = znchar(Mod(139,845))
| Modulus: | \(845\) | |
| Conductor: | \(845\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(78\) |
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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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\(\chi_{845}(9,\cdot)\)
\(\chi_{845}(29,\cdot)\)
\(\chi_{845}(74,\cdot)\)
\(\chi_{845}(94,\cdot)\)
\(\chi_{845}(139,\cdot)\)
\(\chi_{845}(159,\cdot)\)
\(\chi_{845}(204,\cdot)\)
\(\chi_{845}(224,\cdot)\)
\(\chi_{845}(269,\cdot)\)
\(\chi_{845}(289,\cdot)\)
\(\chi_{845}(334,\cdot)\)
\(\chi_{845}(354,\cdot)\)
\(\chi_{845}(399,\cdot)\)
\(\chi_{845}(419,\cdot)\)
\(\chi_{845}(464,\cdot)\)
\(\chi_{845}(549,\cdot)\)
\(\chi_{845}(594,\cdot)\)
\(\chi_{845}(614,\cdot)\)
\(\chi_{845}(659,\cdot)\)
\(\chi_{845}(679,\cdot)\)
\(\chi_{845}(724,\cdot)\)
\(\chi_{845}(744,\cdot)\)
\(\chi_{845}(789,\cdot)\)
\(\chi_{845}(809,\cdot)\)
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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 ]
\((677,171)\) → \((-1,e\left(\frac{14}{39}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 845 }(139, a) \) |
\(1\) | \(1\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{10}{13}\right)\) |
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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)
