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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1081, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([23,36]))
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pari:[g,chi] = znchar(Mod(850,1081))
| Modulus: | \(1081\) | |
| Conductor: | \(1081\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(46\) |
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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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\(\chi_{1081}(68,\cdot)\)
\(\chi_{1081}(183,\cdot)\)
\(\chi_{1081}(206,\cdot)\)
\(\chi_{1081}(252,\cdot)\)
\(\chi_{1081}(298,\cdot)\)
\(\chi_{1081}(390,\cdot)\)
\(\chi_{1081}(413,\cdot)\)
\(\chi_{1081}(459,\cdot)\)
\(\chi_{1081}(482,\cdot)\)
\(\chi_{1081}(551,\cdot)\)
\(\chi_{1081}(620,\cdot)\)
\(\chi_{1081}(643,\cdot)\)
\(\chi_{1081}(666,\cdot)\)
\(\chi_{1081}(712,\cdot)\)
\(\chi_{1081}(758,\cdot)\)
\(\chi_{1081}(827,\cdot)\)
\(\chi_{1081}(850,\cdot)\)
\(\chi_{1081}(873,\cdot)\)
\(\chi_{1081}(896,\cdot)\)
\(\chi_{1081}(942,\cdot)\)
\(\chi_{1081}(965,\cdot)\)
\(\chi_{1081}(1011,\cdot)\)
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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 ]
\((189,898)\) → \((-1,e\left(\frac{18}{23}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 1081 }(850, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{13}{46}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{25}{46}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{45}{46}\right)\) |
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sage:chi.jacobi_sum(n)
