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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2565, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([11,9,2]))
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pari:[g,chi] = znchar(Mod(1049,2565))
| Modulus: | \(2565\) | |
| Conductor: | \(2565\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(18\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{2565}(329,\cdot)\)
\(\chi_{2565}(959,\cdot)\)
\(\chi_{2565}(1049,\cdot)\)
\(\chi_{2565}(1784,\cdot)\)
\(\chi_{2565}(1829,\cdot)\)
\(\chi_{2565}(2144,\cdot)\)
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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 ]
\((191,1027,1351)\) → \((e\left(\frac{11}{18}\right),-1,e\left(\frac{1}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(22\) |
| \( \chi_{ 2565 }(1049, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(-1\) |
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sage:chi.jacobi_sum(n)
