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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1480, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,18,9,2]))
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pari:[g,chi] = znchar(Mod(707,1480))
| Modulus: | \(1480\) | |
| Conductor: | \(1480\) |
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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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{1480}(3,\cdot)\)
\(\chi_{1480}(67,\cdot)\)
\(\chi_{1480}(243,\cdot)\)
\(\chi_{1480}(363,\cdot)\)
\(\chi_{1480}(707,\cdot)\)
\(\chi_{1480}(987,\cdot)\)
\(\chi_{1480}(1003,\cdot)\)
\(\chi_{1480}(1027,\cdot)\)
\(\chi_{1480}(1187,\cdot)\)
\(\chi_{1480}(1283,\cdot)\)
\(\chi_{1480}(1323,\cdot)\)
\(\chi_{1480}(1427,\cdot)\)
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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 ]
\((1111,741,297,1001)\) → \((-1,-1,i,e\left(\frac{1}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 1480 }(707, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) |
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sage:chi.jacobi_sum(n)
