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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([0,18,9,11]))
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pari:[g,chi] = znchar(Mod(1197,1480))
| Modulus: | \(1480\) | |
| Conductor: | \(1480\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(36\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
|
| 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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\(\chi_{1480}(533,\cdot)\)
\(\chi_{1480}(573,\cdot)\)
\(\chi_{1480}(597,\cdot)\)
\(\chi_{1480}(653,\cdot)\)
\(\chi_{1480}(757,\cdot)\)
\(\chi_{1480}(853,\cdot)\)
\(\chi_{1480}(997,\cdot)\)
\(\chi_{1480}(1093,\cdot)\)
\(\chi_{1480}(1197,\cdot)\)
\(\chi_{1480}(1253,\cdot)\)
\(\chi_{1480}(1277,\cdot)\)
\(\chi_{1480}(1317,\cdot)\)
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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 ]
\((1111,741,297,1001)\) → \((1,-1,i,e\left(\frac{11}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 1480 }(1197, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{12}\right)\) |
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sage:chi.jacobi_sum(n)
