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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(345, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,11,15]))
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pari:[g,chi] = znchar(Mod(134,345))
| Modulus: | \(345\) | |
| Conductor: | \(345\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(22\) |
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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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\(\chi_{345}(14,\cdot)\)
\(\chi_{345}(44,\cdot)\)
\(\chi_{345}(74,\cdot)\)
\(\chi_{345}(89,\cdot)\)
\(\chi_{345}(134,\cdot)\)
\(\chi_{345}(149,\cdot)\)
\(\chi_{345}(194,\cdot)\)
\(\chi_{345}(224,\cdot)\)
\(\chi_{345}(314,\cdot)\)
\(\chi_{345}(329,\cdot)\)
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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 ]
\((116,277,166)\) → \((-1,-1,e\left(\frac{15}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 345 }(134, a) \) |
\(1\) | \(1\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) |
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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)
