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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(676, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([39,61]))
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pari:[g,chi] = znchar(Mod(43,676))
| Modulus: | \(676\) | |
| Conductor: | \(676\) |
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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
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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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\(\chi_{676}(43,\cdot)\)
\(\chi_{676}(75,\cdot)\)
\(\chi_{676}(95,\cdot)\)
\(\chi_{676}(127,\cdot)\)
\(\chi_{676}(179,\cdot)\)
\(\chi_{676}(199,\cdot)\)
\(\chi_{676}(231,\cdot)\)
\(\chi_{676}(251,\cdot)\)
\(\chi_{676}(283,\cdot)\)
\(\chi_{676}(303,\cdot)\)
\(\chi_{676}(335,\cdot)\)
\(\chi_{676}(355,\cdot)\)
\(\chi_{676}(387,\cdot)\)
\(\chi_{676}(407,\cdot)\)
\(\chi_{676}(439,\cdot)\)
\(\chi_{676}(459,\cdot)\)
\(\chi_{676}(491,\cdot)\)
\(\chi_{676}(511,\cdot)\)
\(\chi_{676}(543,\cdot)\)
\(\chi_{676}(563,\cdot)\)
\(\chi_{676}(595,\cdot)\)
\(\chi_{676}(615,\cdot)\)
\(\chi_{676}(647,\cdot)\)
\(\chi_{676}(667,\cdot)\)
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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 ]
\((339,509)\) → \((-1,e\left(\frac{61}{78}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 676 }(43, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{1}{6}\right)\) |
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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)
