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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(324, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,52]))
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pari:[g,chi] = znchar(Mod(223,324))
| Modulus: | \(324\) | |
| Conductor: | \(324\) |
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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)
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{324}(7,\cdot)\)
\(\chi_{324}(31,\cdot)\)
\(\chi_{324}(43,\cdot)\)
\(\chi_{324}(67,\cdot)\)
\(\chi_{324}(79,\cdot)\)
\(\chi_{324}(103,\cdot)\)
\(\chi_{324}(115,\cdot)\)
\(\chi_{324}(139,\cdot)\)
\(\chi_{324}(151,\cdot)\)
\(\chi_{324}(175,\cdot)\)
\(\chi_{324}(187,\cdot)\)
\(\chi_{324}(211,\cdot)\)
\(\chi_{324}(223,\cdot)\)
\(\chi_{324}(247,\cdot)\)
\(\chi_{324}(259,\cdot)\)
\(\chi_{324}(283,\cdot)\)
\(\chi_{324}(295,\cdot)\)
\(\chi_{324}(319,\cdot)\)
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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 ]
\((163,245)\) → \((-1,e\left(\frac{26}{27}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 324 }(223, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{41}{54}\right)\) |
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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)
