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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([0,9,16,16]))
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pari:[g,chi] = znchar(Mod(3901,4032))
| 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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{4032}(277,\cdot)\)
\(\chi_{4032}(373,\cdot)\)
\(\chi_{4032}(781,\cdot)\)
\(\chi_{4032}(877,\cdot)\)
\(\chi_{4032}(1285,\cdot)\)
\(\chi_{4032}(1381,\cdot)\)
\(\chi_{4032}(1789,\cdot)\)
\(\chi_{4032}(1885,\cdot)\)
\(\chi_{4032}(2293,\cdot)\)
\(\chi_{4032}(2389,\cdot)\)
\(\chi_{4032}(2797,\cdot)\)
\(\chi_{4032}(2893,\cdot)\)
\(\chi_{4032}(3301,\cdot)\)
\(\chi_{4032}(3397,\cdot)\)
\(\chi_{4032}(3805,\cdot)\)
\(\chi_{4032}(3901,\cdot)\)
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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 ]
\((127,3781,1793,577)\) → \((1,e\left(\frac{3}{16}\right),e\left(\frac{1}{3}\right),e\left(\frac{1}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 4032 }(3901, a) \) |
\(1\) | \(1\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(-1\) | \(e\left(\frac{17}{48}\right)\) |
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sage:chi.jacobi_sum(n)
