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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(128, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,15]))
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pari:[g,chi] = znchar(Mod(115,128))
| Modulus: | \(128\) | |
| Conductor: | \(128\) |
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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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\(\chi_{128}(3,\cdot)\)
\(\chi_{128}(11,\cdot)\)
\(\chi_{128}(19,\cdot)\)
\(\chi_{128}(27,\cdot)\)
\(\chi_{128}(35,\cdot)\)
\(\chi_{128}(43,\cdot)\)
\(\chi_{128}(51,\cdot)\)
\(\chi_{128}(59,\cdot)\)
\(\chi_{128}(67,\cdot)\)
\(\chi_{128}(75,\cdot)\)
\(\chi_{128}(83,\cdot)\)
\(\chi_{128}(91,\cdot)\)
\(\chi_{128}(99,\cdot)\)
\(\chi_{128}(107,\cdot)\)
\(\chi_{128}(115,\cdot)\)
\(\chi_{128}(123,\cdot)\)
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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 ]
\((127,5)\) → \((-1,e\left(\frac{15}{32}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 128 }(115, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{3}{32}\right)\) |
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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)
