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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(231091, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([14,23]))
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pari:[g,chi] = znchar(Mod(198270,231091))
| Modulus: | \(231091\) | |
| Conductor: | \(231091\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(42\) |
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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
|
| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{231091}(31950,\cdot)\)
\(\chi_{231091}(47765,\cdot)\)
\(\chi_{231091}(58662,\cdot)\)
\(\chi_{231091}(60447,\cdot)\)
\(\chi_{231091}(159394,\cdot)\)
\(\chi_{231091}(159644,\cdot)\)
\(\chi_{231091}(161214,\cdot)\)
\(\chi_{231091}(190551,\cdot)\)
\(\chi_{231091}(198270,\cdot)\)
\(\chi_{231091}(201728,\cdot)\)
\(\chi_{231091}(204096,\cdot)\)
\(\chi_{231091}(209941,\cdot)\)
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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 ]
\((66027,99044)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{23}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 231091 }(198270, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(-1\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{11}{21}\right)\) |
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sage:chi.jacobi_sum(n)
