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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(567, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([37,45]))
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pari:[g,chi] = znchar(Mod(110,567))
| Modulus: | \(567\) | |
| Conductor: | \(567\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(54\) |
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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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\(\chi_{567}(47,\cdot)\)
\(\chi_{567}(59,\cdot)\)
\(\chi_{567}(110,\cdot)\)
\(\chi_{567}(122,\cdot)\)
\(\chi_{567}(173,\cdot)\)
\(\chi_{567}(185,\cdot)\)
\(\chi_{567}(236,\cdot)\)
\(\chi_{567}(248,\cdot)\)
\(\chi_{567}(299,\cdot)\)
\(\chi_{567}(311,\cdot)\)
\(\chi_{567}(362,\cdot)\)
\(\chi_{567}(374,\cdot)\)
\(\chi_{567}(425,\cdot)\)
\(\chi_{567}(437,\cdot)\)
\(\chi_{567}(488,\cdot)\)
\(\chi_{567}(500,\cdot)\)
\(\chi_{567}(551,\cdot)\)
\(\chi_{567}(563,\cdot)\)
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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 ]
\((407,325)\) → \((e\left(\frac{37}{54}\right),e\left(\frac{5}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 567 }(110, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{18}\right)\) |
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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)
