from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1340, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,0,59]))
pari: [g,chi] = znchar(Mod(681,1340))
Basic properties
Modulus: | \(1340\) | |
Conductor: | \(67\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{67}(11,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1340.bo
\(\chi_{1340}(41,\cdot)\) \(\chi_{1340}(61,\cdot)\) \(\chi_{1340}(101,\cdot)\) \(\chi_{1340}(141,\cdot)\) \(\chi_{1340}(221,\cdot)\) \(\chi_{1340}(281,\cdot)\) \(\chi_{1340}(381,\cdot)\) \(\chi_{1340}(481,\cdot)\) \(\chi_{1340}(501,\cdot)\) \(\chi_{1340}(621,\cdot)\) \(\chi_{1340}(681,\cdot)\) \(\chi_{1340}(701,\cdot)\) \(\chi_{1340}(721,\cdot)\) \(\chi_{1340}(781,\cdot)\) \(\chi_{1340}(861,\cdot)\) \(\chi_{1340}(921,\cdot)\) \(\chi_{1340}(1001,\cdot)\) \(\chi_{1340}(1141,\cdot)\) \(\chi_{1340}(1301,\cdot)\) \(\chi_{1340}(1321,\cdot)\)
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
\((671,537,1141)\) → \((1,1,e\left(\frac{59}{66}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 1340 }(681, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{13}{22}\right)\) |
sage: chi.jacobi_sum(n)