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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2496, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,33,24,4]))
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pari:[g,chi] = znchar(Mod(1757,2496))
| Modulus: | \(2496\) | |
| Conductor: | \(2496\) |
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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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\(\chi_{2496}(245,\cdot)\)
\(\chi_{2496}(293,\cdot)\)
\(\chi_{2496}(461,\cdot)\)
\(\chi_{2496}(509,\cdot)\)
\(\chi_{2496}(869,\cdot)\)
\(\chi_{2496}(917,\cdot)\)
\(\chi_{2496}(1085,\cdot)\)
\(\chi_{2496}(1133,\cdot)\)
\(\chi_{2496}(1493,\cdot)\)
\(\chi_{2496}(1541,\cdot)\)
\(\chi_{2496}(1709,\cdot)\)
\(\chi_{2496}(1757,\cdot)\)
\(\chi_{2496}(2117,\cdot)\)
\(\chi_{2496}(2165,\cdot)\)
\(\chi_{2496}(2333,\cdot)\)
\(\chi_{2496}(2381,\cdot)\)
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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 ]
\((703,1093,833,769)\) → \((1,e\left(\frac{11}{16}\right),-1,e\left(\frac{1}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 2496 }(1757, a) \) |
\(1\) | \(1\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(i\) | \(e\left(\frac{35}{48}\right)\) |
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sage:chi.jacobi_sum(n)
