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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2160, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,27,16,27]))
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pari:[g,chi] = znchar(Mod(1363,2160))
| Modulus: | \(2160\) | |
| Conductor: | \(2160\) |
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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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\(\chi_{2160}(187,\cdot)\)
\(\chi_{2160}(403,\cdot)\)
\(\chi_{2160}(427,\cdot)\)
\(\chi_{2160}(643,\cdot)\)
\(\chi_{2160}(907,\cdot)\)
\(\chi_{2160}(1123,\cdot)\)
\(\chi_{2160}(1147,\cdot)\)
\(\chi_{2160}(1363,\cdot)\)
\(\chi_{2160}(1627,\cdot)\)
\(\chi_{2160}(1843,\cdot)\)
\(\chi_{2160}(1867,\cdot)\)
\(\chi_{2160}(2083,\cdot)\)
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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 ]
\((271,1621,2081,1297)\) → \((-1,-i,e\left(\frac{4}{9}\right),-i)\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 2160 }(1363, a) \) |
\(1\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{18}\right)\) |
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sage:chi.jacobi_sum(n)
