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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(297, base_ring=CyclotomicField(90))
M = H._module
chi = DirichletCharacter(H, M([85,81]))
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pari:[g,chi] = znchar(Mod(149,297))
| Modulus: | \(297\) | |
| Conductor: | \(297\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(90\) |
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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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{297}(2,\cdot)\)
\(\chi_{297}(29,\cdot)\)
\(\chi_{297}(41,\cdot)\)
\(\chi_{297}(50,\cdot)\)
\(\chi_{297}(68,\cdot)\)
\(\chi_{297}(74,\cdot)\)
\(\chi_{297}(83,\cdot)\)
\(\chi_{297}(95,\cdot)\)
\(\chi_{297}(101,\cdot)\)
\(\chi_{297}(128,\cdot)\)
\(\chi_{297}(140,\cdot)\)
\(\chi_{297}(149,\cdot)\)
\(\chi_{297}(167,\cdot)\)
\(\chi_{297}(173,\cdot)\)
\(\chi_{297}(182,\cdot)\)
\(\chi_{297}(194,\cdot)\)
\(\chi_{297}(200,\cdot)\)
\(\chi_{297}(227,\cdot)\)
\(\chi_{297}(239,\cdot)\)
\(\chi_{297}(248,\cdot)\)
\(\chi_{297}(266,\cdot)\)
\(\chi_{297}(272,\cdot)\)
\(\chi_{297}(281,\cdot)\)
\(\chi_{297}(293,\cdot)\)
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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 ]
\((56,244)\) → \((e\left(\frac{17}{18}\right),e\left(\frac{9}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 297 }(149, a) \) |
\(1\) | \(1\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{29}{90}\right)\) | \(e\left(\frac{37}{90}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{41}{90}\right)\) | \(e\left(\frac{23}{90}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{4}{15}\right)\) |
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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)
