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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2268, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,50,9]))
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pari:[g,chi] = znchar(Mod(1291,2268))
| Modulus: | \(2268\) | |
| Conductor: | \(2268\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| 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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\(\chi_{2268}(31,\cdot)\)
\(\chi_{2268}(187,\cdot)\)
\(\chi_{2268}(283,\cdot)\)
\(\chi_{2268}(439,\cdot)\)
\(\chi_{2268}(535,\cdot)\)
\(\chi_{2268}(691,\cdot)\)
\(\chi_{2268}(787,\cdot)\)
\(\chi_{2268}(943,\cdot)\)
\(\chi_{2268}(1039,\cdot)\)
\(\chi_{2268}(1195,\cdot)\)
\(\chi_{2268}(1291,\cdot)\)
\(\chi_{2268}(1447,\cdot)\)
\(\chi_{2268}(1543,\cdot)\)
\(\chi_{2268}(1699,\cdot)\)
\(\chi_{2268}(1795,\cdot)\)
\(\chi_{2268}(1951,\cdot)\)
\(\chi_{2268}(2047,\cdot)\)
\(\chi_{2268}(2203,\cdot)\)
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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 ]
\((1135,1541,325)\) → \((-1,e\left(\frac{25}{27}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 2268 }(1291, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{2}{9}\right)\) |
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sage:chi.jacobi_sum(n)
