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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([27,22,9]))
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pari:[g,chi] = znchar(Mod(367,2268))
| Modulus: | \(2268\) | |
| Conductor: | \(2268\) |
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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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\(\chi_{2268}(103,\cdot)\)
\(\chi_{2268}(115,\cdot)\)
\(\chi_{2268}(355,\cdot)\)
\(\chi_{2268}(367,\cdot)\)
\(\chi_{2268}(607,\cdot)\)
\(\chi_{2268}(619,\cdot)\)
\(\chi_{2268}(859,\cdot)\)
\(\chi_{2268}(871,\cdot)\)
\(\chi_{2268}(1111,\cdot)\)
\(\chi_{2268}(1123,\cdot)\)
\(\chi_{2268}(1363,\cdot)\)
\(\chi_{2268}(1375,\cdot)\)
\(\chi_{2268}(1615,\cdot)\)
\(\chi_{2268}(1627,\cdot)\)
\(\chi_{2268}(1867,\cdot)\)
\(\chi_{2268}(1879,\cdot)\)
\(\chi_{2268}(2119,\cdot)\)
\(\chi_{2268}(2131,\cdot)\)
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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 ]
\((1135,1541,325)\) → \((-1,e\left(\frac{11}{27}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 2268 }(367, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{4}{9}\right)\) |
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sage:chi.jacobi_sum(n)
