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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(576, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,9,16]))
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pari:[g,chi] = znchar(Mod(445,576))
| Modulus: | \(576\) | |
| Conductor: | \(576\) |
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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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\(\chi_{576}(13,\cdot)\)
\(\chi_{576}(61,\cdot)\)
\(\chi_{576}(85,\cdot)\)
\(\chi_{576}(133,\cdot)\)
\(\chi_{576}(157,\cdot)\)
\(\chi_{576}(205,\cdot)\)
\(\chi_{576}(229,\cdot)\)
\(\chi_{576}(277,\cdot)\)
\(\chi_{576}(301,\cdot)\)
\(\chi_{576}(349,\cdot)\)
\(\chi_{576}(373,\cdot)\)
\(\chi_{576}(421,\cdot)\)
\(\chi_{576}(445,\cdot)\)
\(\chi_{576}(493,\cdot)\)
\(\chi_{576}(517,\cdot)\)
\(\chi_{576}(565,\cdot)\)
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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 ]
\((127,325,65)\) → \((1,e\left(\frac{3}{16}\right),e\left(\frac{1}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 576 }(445, a) \) |
\(1\) | \(1\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(i\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) |
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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)
