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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(384, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,27,16]))
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pari:[g,chi] = znchar(Mod(29,384))
| Modulus: | \(384\) | |
| Conductor: | \(384\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(32\) |
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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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\(\chi_{384}(5,\cdot)\)
\(\chi_{384}(29,\cdot)\)
\(\chi_{384}(53,\cdot)\)
\(\chi_{384}(77,\cdot)\)
\(\chi_{384}(101,\cdot)\)
\(\chi_{384}(125,\cdot)\)
\(\chi_{384}(149,\cdot)\)
\(\chi_{384}(173,\cdot)\)
\(\chi_{384}(197,\cdot)\)
\(\chi_{384}(221,\cdot)\)
\(\chi_{384}(245,\cdot)\)
\(\chi_{384}(269,\cdot)\)
\(\chi_{384}(293,\cdot)\)
\(\chi_{384}(317,\cdot)\)
\(\chi_{384}(341,\cdot)\)
\(\chi_{384}(365,\cdot)\)
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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 ]
\((127,133,257)\) → \((1,e\left(\frac{27}{32}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 384 }(29, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(-i\) |
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sage:chi.jacobi_sum(n)
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sage:chi.gauss_sum(a)
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pari:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
