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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(676, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([78,55]))
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pari:[g,chi] = znchar(Mod(323,676))
| Modulus: | \(676\) | |
| Conductor: | \(676\) |
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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)
|
\(\chi_{676}(7,\cdot)\)
\(\chi_{676}(11,\cdot)\)
\(\chi_{676}(15,\cdot)\)
\(\chi_{676}(59,\cdot)\)
\(\chi_{676}(63,\cdot)\)
\(\chi_{676}(67,\cdot)\)
\(\chi_{676}(71,\cdot)\)
\(\chi_{676}(111,\cdot)\)
\(\chi_{676}(115,\cdot)\)
\(\chi_{676}(119,\cdot)\)
\(\chi_{676}(123,\cdot)\)
\(\chi_{676}(163,\cdot)\)
\(\chi_{676}(167,\cdot)\)
\(\chi_{676}(171,\cdot)\)
\(\chi_{676}(175,\cdot)\)
\(\chi_{676}(215,\cdot)\)
\(\chi_{676}(219,\cdot)\)
\(\chi_{676}(223,\cdot)\)
\(\chi_{676}(227,\cdot)\)
\(\chi_{676}(267,\cdot)\)
\(\chi_{676}(271,\cdot)\)
\(\chi_{676}(275,\cdot)\)
\(\chi_{676}(279,\cdot)\)
\(\chi_{676}(323,\cdot)\)
\(\chi_{676}(327,\cdot)\)
\(\chi_{676}(331,\cdot)\)
\(\chi_{676}(371,\cdot)\)
\(\chi_{676}(375,\cdot)\)
\(\chi_{676}(379,\cdot)\)
\(\chi_{676}(383,\cdot)\)
...
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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 ]
\((339,509)\) → \((-1,e\left(\frac{55}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 676 }(323, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{78}\right)\) | \(e\left(\frac{9}{52}\right)\) | \(e\left(\frac{35}{156}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{127}{156}\right)\) | \(e\left(\frac{61}{156}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{23}{52}\right)\) | \(e\left(\frac{1}{3}\right)\) |
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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)
