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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2856, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,24,24,32,9]))
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pari:[g,chi] = znchar(Mod(2237,2856))
| Modulus: | \(2856\) | |
| Conductor: | \(2856\) |
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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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\(\chi_{2856}(317,\cdot)\)
\(\chi_{2856}(653,\cdot)\)
\(\chi_{2856}(725,\cdot)\)
\(\chi_{2856}(821,\cdot)\)
\(\chi_{2856}(989,\cdot)\)
\(\chi_{2856}(1061,\cdot)\)
\(\chi_{2856}(1229,\cdot)\)
\(\chi_{2856}(1397,\cdot)\)
\(\chi_{2856}(1493,\cdot)\)
\(\chi_{2856}(1661,\cdot)\)
\(\chi_{2856}(1829,\cdot)\)
\(\chi_{2856}(1901,\cdot)\)
\(\chi_{2856}(2069,\cdot)\)
\(\chi_{2856}(2165,\cdot)\)
\(\chi_{2856}(2237,\cdot)\)
\(\chi_{2856}(2573,\cdot)\)
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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 ]
\((2143,1429,953,409,2689)\) → \((1,-1,-1,e\left(\frac{2}{3}\right),e\left(\frac{3}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 2856 }(2237, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(i\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{9}{16}\right)\) |
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sage:chi.jacobi_sum(n)
