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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(465, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,15,22]))
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pari:[g,chi] = znchar(Mod(137,465))
| Modulus: | \(465\) | |
| Conductor: | \(465\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(60\) |
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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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\(\chi_{465}(17,\cdot)\)
\(\chi_{465}(53,\cdot)\)
\(\chi_{465}(83,\cdot)\)
\(\chi_{465}(137,\cdot)\)
\(\chi_{465}(158,\cdot)\)
\(\chi_{465}(167,\cdot)\)
\(\chi_{465}(197,\cdot)\)
\(\chi_{465}(203,\cdot)\)
\(\chi_{465}(272,\cdot)\)
\(\chi_{465}(323,\cdot)\)
\(\chi_{465}(332,\cdot)\)
\(\chi_{465}(353,\cdot)\)
\(\chi_{465}(362,\cdot)\)
\(\chi_{465}(383,\cdot)\)
\(\chi_{465}(437,\cdot)\)
\(\chi_{465}(458,\cdot)\)
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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 ]
\((311,187,406)\) → \((-1,i,e\left(\frac{11}{30}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 465 }(137, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) |
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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)
