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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4032, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,45,16,8]))
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pari:[g,chi] = znchar(Mod(1165,4032))
| Modulus: | \(4032\) | |
| Conductor: | \(4032\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(48\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
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| 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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\(\chi_{4032}(61,\cdot)\)
\(\chi_{4032}(157,\cdot)\)
\(\chi_{4032}(565,\cdot)\)
\(\chi_{4032}(661,\cdot)\)
\(\chi_{4032}(1069,\cdot)\)
\(\chi_{4032}(1165,\cdot)\)
\(\chi_{4032}(1573,\cdot)\)
\(\chi_{4032}(1669,\cdot)\)
\(\chi_{4032}(2077,\cdot)\)
\(\chi_{4032}(2173,\cdot)\)
\(\chi_{4032}(2581,\cdot)\)
\(\chi_{4032}(2677,\cdot)\)
\(\chi_{4032}(3085,\cdot)\)
\(\chi_{4032}(3181,\cdot)\)
\(\chi_{4032}(3589,\cdot)\)
\(\chi_{4032}(3685,\cdot)\)
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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 ]
\((127,3781,1793,577)\) → \((1,e\left(\frac{15}{16}\right),e\left(\frac{1}{3}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 4032 }(1165, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{37}{48}\right)\) |
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sage:chi.jacobi_sum(n)
