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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([117,122]))
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pari:[g,chi] = znchar(Mod(43,845))
| 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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
|
\(\chi_{845}(17,\cdot)\)
\(\chi_{845}(43,\cdot)\)
\(\chi_{845}(62,\cdot)\)
\(\chi_{845}(82,\cdot)\)
\(\chi_{845}(88,\cdot)\)
\(\chi_{845}(108,\cdot)\)
\(\chi_{845}(127,\cdot)\)
\(\chi_{845}(153,\cdot)\)
\(\chi_{845}(173,\cdot)\)
\(\chi_{845}(212,\cdot)\)
\(\chi_{845}(218,\cdot)\)
\(\chi_{845}(238,\cdot)\)
\(\chi_{845}(257,\cdot)\)
\(\chi_{845}(277,\cdot)\)
\(\chi_{845}(283,\cdot)\)
\(\chi_{845}(303,\cdot)\)
\(\chi_{845}(322,\cdot)\)
\(\chi_{845}(342,\cdot)\)
\(\chi_{845}(348,\cdot)\)
\(\chi_{845}(368,\cdot)\)
\(\chi_{845}(387,\cdot)\)
\(\chi_{845}(407,\cdot)\)
\(\chi_{845}(413,\cdot)\)
\(\chi_{845}(433,\cdot)\)
\(\chi_{845}(452,\cdot)\)
\(\chi_{845}(472,\cdot)\)
\(\chi_{845}(478,\cdot)\)
\(\chi_{845}(498,\cdot)\)
\(\chi_{845}(517,\cdot)\)
\(\chi_{845}(537,\cdot)\)
...
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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 ]
\((677,171)\) → \((-i,e\left(\frac{61}{78}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 845 }(43, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{83}{156}\right)\) | \(e\left(\frac{35}{156}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{67}{156}\right)\) | \(e\left(\frac{31}{52}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{15}{52}\right)\) | \(e\left(\frac{25}{26}\right)\) |
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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)
