from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1005, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,25]))
pari: [g,chi] = znchar(Mod(899,1005))
Basic properties
Modulus: | \(1005\) | |
Conductor: | \(1005\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1005.bo
\(\chi_{1005}(44,\cdot)\) \(\chi_{1005}(74,\cdot)\) \(\chi_{1005}(299,\cdot)\) \(\chi_{1005}(314,\cdot)\) \(\chi_{1005}(329,\cdot)\) \(\chi_{1005}(404,\cdot)\) \(\chi_{1005}(434,\cdot)\) \(\chi_{1005}(554,\cdot)\) \(\chi_{1005}(584,\cdot)\) \(\chi_{1005}(599,\cdot)\) \(\chi_{1005}(614,\cdot)\) \(\chi_{1005}(644,\cdot)\) \(\chi_{1005}(704,\cdot)\) \(\chi_{1005}(749,\cdot)\) \(\chi_{1005}(794,\cdot)\) \(\chi_{1005}(824,\cdot)\) \(\chi_{1005}(854,\cdot)\) \(\chi_{1005}(884,\cdot)\) \(\chi_{1005}(899,\cdot)\) \(\chi_{1005}(989,\cdot)\)
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
\((671,202,136)\) → \((-1,-1,e\left(\frac{25}{66}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 1005 }(899, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) |
sage: chi.jacobi_sum(n)