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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([27,29,36]))
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pari:[g,chi] = znchar(Mod(1859,2268))
| 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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\(\chi_{2268}(95,\cdot)\)
\(\chi_{2268}(191,\cdot)\)
\(\chi_{2268}(347,\cdot)\)
\(\chi_{2268}(443,\cdot)\)
\(\chi_{2268}(599,\cdot)\)
\(\chi_{2268}(695,\cdot)\)
\(\chi_{2268}(851,\cdot)\)
\(\chi_{2268}(947,\cdot)\)
\(\chi_{2268}(1103,\cdot)\)
\(\chi_{2268}(1199,\cdot)\)
\(\chi_{2268}(1355,\cdot)\)
\(\chi_{2268}(1451,\cdot)\)
\(\chi_{2268}(1607,\cdot)\)
\(\chi_{2268}(1703,\cdot)\)
\(\chi_{2268}(1859,\cdot)\)
\(\chi_{2268}(1955,\cdot)\)
\(\chi_{2268}(2111,\cdot)\)
\(\chi_{2268}(2207,\cdot)\)
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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 ]
\((1135,1541,325)\) → \((-1,e\left(\frac{29}{54}\right),e\left(\frac{2}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 2268 }(1859, a) \) |
\(1\) | \(1\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{8}{9}\right)\) |
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sage:chi.jacobi_sum(n)
