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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5200, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,45,36,35]))
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pari:[g,chi] = znchar(Mod(1571,5200))
| Modulus: | \(5200\) | |
| Conductor: | \(5200\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(60\) |
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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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\(\chi_{5200}(171,\cdot)\)
\(\chi_{5200}(331,\cdot)\)
\(\chi_{5200}(371,\cdot)\)
\(\chi_{5200}(531,\cdot)\)
\(\chi_{5200}(1211,\cdot)\)
\(\chi_{5200}(1371,\cdot)\)
\(\chi_{5200}(1411,\cdot)\)
\(\chi_{5200}(1571,\cdot)\)
\(\chi_{5200}(2411,\cdot)\)
\(\chi_{5200}(2611,\cdot)\)
\(\chi_{5200}(3291,\cdot)\)
\(\chi_{5200}(3491,\cdot)\)
\(\chi_{5200}(4331,\cdot)\)
\(\chi_{5200}(4491,\cdot)\)
\(\chi_{5200}(4531,\cdot)\)
\(\chi_{5200}(4691,\cdot)\)
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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 ]
\((1951,1301,4577,1601)\) → \((-1,-i,e\left(\frac{3}{5}\right),e\left(\frac{7}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 5200 }(1571, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{47}{60}\right)\) |
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sage:chi.jacobi_sum(n)
