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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(760, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,18,27,2]))
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pari:[g,chi] = znchar(Mod(363,760))
| Modulus: | \(760\) | |
| Conductor: | \(760\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(36\) |
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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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\(\chi_{760}(3,\cdot)\)
\(\chi_{760}(67,\cdot)\)
\(\chi_{760}(147,\cdot)\)
\(\chi_{760}(203,\cdot)\)
\(\chi_{760}(243,\cdot)\)
\(\chi_{760}(307,\cdot)\)
\(\chi_{760}(363,\cdot)\)
\(\chi_{760}(507,\cdot)\)
\(\chi_{760}(523,\cdot)\)
\(\chi_{760}(547,\cdot)\)
\(\chi_{760}(603,\cdot)\)
\(\chi_{760}(667,\cdot)\)
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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 ]
\((191,381,457,401)\) → \((-1,-1,-i,e\left(\frac{1}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 760 }(363, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{17}{18}\right)\) |
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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)
