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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1588, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,25]))
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pari:[g,chi] = znchar(Mod(1103,1588))
| Modulus: | \(1588\) | |
| Conductor: | \(1588\) |
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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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\(\chi_{1588}(203,\cdot)\)
\(\chi_{1588}(591,\cdot)\)
\(\chi_{1588}(835,\cdot)\)
\(\chi_{1588}(947,\cdot)\)
\(\chi_{1588}(1007,\cdot)\)
\(\chi_{1588}(1095,\cdot)\)
\(\chi_{1588}(1103,\cdot)\)
\(\chi_{1588}(1279,\cdot)\)
\(\chi_{1588}(1287,\cdot)\)
\(\chi_{1588}(1375,\cdot)\)
\(\chi_{1588}(1435,\cdot)\)
\(\chi_{1588}(1547,\cdot)\)
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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 ]
\((795,5)\) → \((-1,e\left(\frac{25}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1588 }(1103, a) \) |
\(1\) | \(1\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{31}{36}\right)\) |
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sage:chi.jacobi_sum(n)
