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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8512, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,27,8,24]))
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pari:[g,chi] = znchar(Mod(3419,8512))
| Modulus: | \(8512\) | |
| Conductor: | \(8512\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(48\) |
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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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\(\chi_{8512}(75,\cdot)\)
\(\chi_{8512}(227,\cdot)\)
\(\chi_{8512}(1139,\cdot)\)
\(\chi_{8512}(1291,\cdot)\)
\(\chi_{8512}(2203,\cdot)\)
\(\chi_{8512}(2355,\cdot)\)
\(\chi_{8512}(3267,\cdot)\)
\(\chi_{8512}(3419,\cdot)\)
\(\chi_{8512}(4331,\cdot)\)
\(\chi_{8512}(4483,\cdot)\)
\(\chi_{8512}(5395,\cdot)\)
\(\chi_{8512}(5547,\cdot)\)
\(\chi_{8512}(6459,\cdot)\)
\(\chi_{8512}(6611,\cdot)\)
\(\chi_{8512}(7523,\cdot)\)
\(\chi_{8512}(7675,\cdot)\)
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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 ]
\((5055,6917,7297,3137)\) → \((-1,e\left(\frac{9}{16}\right),e\left(\frac{1}{6}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 8512 }(3419, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(i\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{9}{16}\right)\) |
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sage:chi.jacobi_sum(n)
