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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5712, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,36,24,32,15]))
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pari:[g,chi] = znchar(Mod(515,5712))
| Modulus: | \(5712\) | |
| Conductor: | \(5712\) |
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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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\(\chi_{5712}(11,\cdot)\)
\(\chi_{5712}(275,\cdot)\)
\(\chi_{5712}(347,\cdot)\)
\(\chi_{5712}(515,\cdot)\)
\(\chi_{5712}(947,\cdot)\)
\(\chi_{5712}(1115,\cdot)\)
\(\chi_{5712}(1187,\cdot)\)
\(\chi_{5712}(1451,\cdot)\)
\(\chi_{5712}(2459,\cdot)\)
\(\chi_{5712}(2795,\cdot)\)
\(\chi_{5712}(2963,\cdot)\)
\(\chi_{5712}(3539,\cdot)\)
\(\chi_{5712}(3635,\cdot)\)
\(\chi_{5712}(4211,\cdot)\)
\(\chi_{5712}(4379,\cdot)\)
\(\chi_{5712}(4715,\cdot)\)
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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 ]
\((2143,1429,3809,3265,2689)\) → \((-1,-i,-1,e\left(\frac{2}{3}\right),e\left(\frac{5}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 5712 }(515, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(-1\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{7}{16}\right)\) |
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sage:chi.jacobi_sum(n)
