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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(368, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,33,26]))
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pari:[g,chi] = znchar(Mod(67,368))
| Modulus: | \(368\) | |
| Conductor: | \(368\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(44\) |
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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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\(\chi_{368}(11,\cdot)\)
\(\chi_{368}(19,\cdot)\)
\(\chi_{368}(43,\cdot)\)
\(\chi_{368}(51,\cdot)\)
\(\chi_{368}(67,\cdot)\)
\(\chi_{368}(83,\cdot)\)
\(\chi_{368}(99,\cdot)\)
\(\chi_{368}(107,\cdot)\)
\(\chi_{368}(155,\cdot)\)
\(\chi_{368}(171,\cdot)\)
\(\chi_{368}(195,\cdot)\)
\(\chi_{368}(203,\cdot)\)
\(\chi_{368}(227,\cdot)\)
\(\chi_{368}(235,\cdot)\)
\(\chi_{368}(251,\cdot)\)
\(\chi_{368}(267,\cdot)\)
\(\chi_{368}(283,\cdot)\)
\(\chi_{368}(291,\cdot)\)
\(\chi_{368}(339,\cdot)\)
\(\chi_{368}(355,\cdot)\)
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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 ]
\((47,277,97)\) → \((-1,-i,e\left(\frac{13}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 368 }(67, a) \) |
\(1\) | \(1\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{19}{44}\right)\) |
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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)
