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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(669, base_ring=CyclotomicField(222))
M = H._module
chi = DirichletCharacter(H, M([111,175]))
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pari:[g,chi] = znchar(Mod(290,669))
| Modulus: | \(669\) | |
| Conductor: | \(669\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(222\) |
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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)
|
\(\chi_{669}(5,\cdot)\)
\(\chi_{669}(11,\cdot)\)
\(\chi_{669}(20,\cdot)\)
\(\chi_{669}(23,\cdot)\)
\(\chi_{669}(35,\cdot)\)
\(\chi_{669}(44,\cdot)\)
\(\chi_{669}(71,\cdot)\)
\(\chi_{669}(77,\cdot)\)
\(\chi_{669}(80,\cdot)\)
\(\chi_{669}(92,\cdot)\)
\(\chi_{669}(107,\cdot)\)
\(\chi_{669}(113,\cdot)\)
\(\chi_{669}(122,\cdot)\)
\(\chi_{669}(134,\cdot)\)
\(\chi_{669}(137,\cdot)\)
\(\chi_{669}(140,\cdot)\)
\(\chi_{669}(149,\cdot)\)
\(\chi_{669}(158,\cdot)\)
\(\chi_{669}(161,\cdot)\)
\(\chi_{669}(170,\cdot)\)
\(\chi_{669}(173,\cdot)\)
\(\chi_{669}(176,\cdot)\)
\(\chi_{669}(185,\cdot)\)
\(\chi_{669}(194,\cdot)\)
\(\chi_{669}(233,\cdot)\)
\(\chi_{669}(245,\cdot)\)
\(\chi_{669}(269,\cdot)\)
\(\chi_{669}(284,\cdot)\)
\(\chi_{669}(290,\cdot)\)
\(\chi_{669}(293,\cdot)\)
...
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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 ]
\((224,226)\) → \((-1,e\left(\frac{175}{222}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 669 }(290, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{74}\right)\) | \(e\left(\frac{29}{37}\right)\) | \(e\left(\frac{73}{111}\right)\) | \(e\left(\frac{20}{37}\right)\) | \(e\left(\frac{13}{74}\right)\) | \(e\left(\frac{11}{222}\right)\) | \(e\left(\frac{94}{111}\right)\) | \(e\left(\frac{65}{74}\right)\) | \(e\left(\frac{69}{74}\right)\) | \(e\left(\frac{21}{37}\right)\) |
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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)
