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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2420, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,11,2]))
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pari:[g,chi] = znchar(Mod(419,2420))
| Modulus: | \(2420\) | |
| Conductor: | \(2420\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(22\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{2420}(199,\cdot)\)
\(\chi_{2420}(419,\cdot)\)
\(\chi_{2420}(639,\cdot)\)
\(\chi_{2420}(859,\cdot)\)
\(\chi_{2420}(1079,\cdot)\)
\(\chi_{2420}(1299,\cdot)\)
\(\chi_{2420}(1519,\cdot)\)
\(\chi_{2420}(1739,\cdot)\)
\(\chi_{2420}(1959,\cdot)\)
\(\chi_{2420}(2399,\cdot)\)
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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 ]
\((1211,1937,2301)\) → \((-1,-1,e\left(\frac{1}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 2420 }(419, a) \) |
\(-1\) | \(1\) | \(1\) | \(e\left(\frac{7}{11}\right)\) | \(1\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(1\) | \(e\left(\frac{6}{11}\right)\) |
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sage:chi.jacobi_sum(n)
