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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(832, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,21,20]))
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pari:[g,chi] = znchar(Mod(45,832))
| Modulus: | \(832\) | |
| Conductor: | \(832\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{832}(37,\cdot)\)
\(\chi_{832}(45,\cdot)\)
\(\chi_{832}(85,\cdot)\)
\(\chi_{832}(93,\cdot)\)
\(\chi_{832}(245,\cdot)\)
\(\chi_{832}(253,\cdot)\)
\(\chi_{832}(293,\cdot)\)
\(\chi_{832}(301,\cdot)\)
\(\chi_{832}(453,\cdot)\)
\(\chi_{832}(461,\cdot)\)
\(\chi_{832}(501,\cdot)\)
\(\chi_{832}(509,\cdot)\)
\(\chi_{832}(661,\cdot)\)
\(\chi_{832}(669,\cdot)\)
\(\chi_{832}(709,\cdot)\)
\(\chi_{832}(717,\cdot)\)
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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 ]
\((703,261,769)\) → \((1,e\left(\frac{7}{16}\right),e\left(\frac{5}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 832 }(45, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{7}{24}\right)\) |
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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)
