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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(11424, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,30,24,40,15]))
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pari:[g,chi] = znchar(Mod(5003,11424))
| Modulus: | \(11424\) | |
| Conductor: | \(11424\) |
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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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{11424}(1235,\cdot)\)
\(\chi_{11424}(1571,\cdot)\)
\(\chi_{11424}(1739,\cdot)\)
\(\chi_{11424}(2987,\cdot)\)
\(\chi_{11424}(4499,\cdot)\)
\(\chi_{11424}(4763,\cdot)\)
\(\chi_{11424}(4835,\cdot)\)
\(\chi_{11424}(5003,\cdot)\)
\(\chi_{11424}(5603,\cdot)\)
\(\chi_{11424}(5939,\cdot)\)
\(\chi_{11424}(8027,\cdot)\)
\(\chi_{11424}(8123,\cdot)\)
\(\chi_{11424}(8867,\cdot)\)
\(\chi_{11424}(9203,\cdot)\)
\(\chi_{11424}(11147,\cdot)\)
\(\chi_{11424}(11387,\cdot)\)
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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 ]
\((2143,7141,3809,3265,2689)\) → \((-1,e\left(\frac{5}{8}\right),-1,e\left(\frac{5}{6}\right),e\left(\frac{5}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 11424 }(5003, a) \) |
\(1\) | \(1\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{3}{16}\right)\) |
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sage:chi.jacobi_sum(n)
