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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2001, base_ring=CyclotomicField(154))
M = H._module
chi = DirichletCharacter(H, M([77,7,33]))
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pari:[g,chi] = znchar(Mod(557,2001))
| Modulus: | \(2001\) | |
| Conductor: | \(2001\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(154\) |
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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)
|
\(\chi_{2001}(5,\cdot)\)
\(\chi_{2001}(38,\cdot)\)
\(\chi_{2001}(80,\cdot)\)
\(\chi_{2001}(122,\cdot)\)
\(\chi_{2001}(125,\cdot)\)
\(\chi_{2001}(149,\cdot)\)
\(\chi_{2001}(158,\cdot)\)
\(\chi_{2001}(212,\cdot)\)
\(\chi_{2001}(245,\cdot)\)
\(\chi_{2001}(296,\cdot)\)
\(\chi_{2001}(332,\cdot)\)
\(\chi_{2001}(341,\cdot)\)
\(\chi_{2001}(383,\cdot)\)
\(\chi_{2001}(410,\cdot)\)
\(\chi_{2001}(419,\cdot)\)
\(\chi_{2001}(428,\cdot)\)
\(\chi_{2001}(470,\cdot)\)
\(\chi_{2001}(497,\cdot)\)
\(\chi_{2001}(527,\cdot)\)
\(\chi_{2001}(557,\cdot)\)
\(\chi_{2001}(701,\cdot)\)
\(\chi_{2001}(734,\cdot)\)
\(\chi_{2001}(776,\cdot)\)
\(\chi_{2001}(845,\cdot)\)
\(\chi_{2001}(908,\cdot)\)
\(\chi_{2001}(941,\cdot)\)
\(\chi_{2001}(950,\cdot)\)
\(\chi_{2001}(962,\cdot)\)
\(\chi_{2001}(1019,\cdot)\)
\(\chi_{2001}(1049,\cdot)\)
...
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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 ]
\((668,1132,553)\) → \((-1,e\left(\frac{1}{22}\right),e\left(\frac{3}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 2001 }(557, a) \) |
\(1\) | \(1\) | \(e\left(\frac{62}{77}\right)\) | \(e\left(\frac{47}{77}\right)\) | \(e\left(\frac{20}{77}\right)\) | \(e\left(\frac{67}{154}\right)\) | \(e\left(\frac{32}{77}\right)\) | \(e\left(\frac{5}{77}\right)\) | \(e\left(\frac{41}{154}\right)\) | \(e\left(\frac{38}{77}\right)\) | \(e\left(\frac{37}{154}\right)\) | \(e\left(\frac{17}{77}\right)\) |
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sage:chi.jacobi_sum(n)
