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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(740, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,18,25]))
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pari:[g,chi] = znchar(Mod(279,740))
| Modulus: | \(740\) | |
| Conductor: | \(740\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(36\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
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| 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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\(\chi_{740}(19,\cdot)\)
\(\chi_{740}(39,\cdot)\)
\(\chi_{740}(59,\cdot)\)
\(\chi_{740}(79,\cdot)\)
\(\chi_{740}(239,\cdot)\)
\(\chi_{740}(279,\cdot)\)
\(\chi_{740}(439,\cdot)\)
\(\chi_{740}(459,\cdot)\)
\(\chi_{740}(479,\cdot)\)
\(\chi_{740}(499,\cdot)\)
\(\chi_{740}(579,\cdot)\)
\(\chi_{740}(679,\cdot)\)
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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 ]
\((371,297,261)\) → \((-1,-1,e\left(\frac{25}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 740 }(279, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) |
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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)
