from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4508, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,11,63]))
pari: [g,chi] = znchar(Mod(129,4508))
Basic properties
Modulus: | \(4508\) | |
Conductor: | \(161\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{161}(129,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4508.bs
\(\chi_{4508}(129,\cdot)\) \(\chi_{4508}(313,\cdot)\) \(\chi_{4508}(521,\cdot)\) \(\chi_{4508}(705,\cdot)\) \(\chi_{4508}(1109,\cdot)\) \(\chi_{4508}(1293,\cdot)\) \(\chi_{4508}(1305,\cdot)\) \(\chi_{4508}(1489,\cdot)\) \(\chi_{4508}(1893,\cdot)\) \(\chi_{4508}(2077,\cdot)\) \(\chi_{4508}(2089,\cdot)\) \(\chi_{4508}(2273,\cdot)\) \(\chi_{4508}(2481,\cdot)\) \(\chi_{4508}(2665,\cdot)\) \(\chi_{4508}(2873,\cdot)\) \(\chi_{4508}(3057,\cdot)\) \(\chi_{4508}(3069,\cdot)\) \(\chi_{4508}(3253,\cdot)\) \(\chi_{4508}(3461,\cdot)\) \(\chi_{4508}(3645,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
Field of values: | \(\Q(\zeta_{33})\) |
Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((2255,1473,1569)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{21}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(25\) | \(27\) |
\( \chi_{ 4508 }(129, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{7}{22}\right)\) |
sage: chi.jacobi_sum(n)