from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1340, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,41]))
pari: [g,chi] = znchar(Mod(79,1340))
Basic properties
| Modulus: | \(1340\) | |
| Conductor: | \(1340\) | 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 1340.br
\(\chi_{1340}(79,\cdot)\) \(\chi_{1340}(99,\cdot)\) \(\chi_{1340}(219,\cdot)\) \(\chi_{1340}(279,\cdot)\) \(\chi_{1340}(299,\cdot)\) \(\chi_{1340}(319,\cdot)\) \(\chi_{1340}(379,\cdot)\) \(\chi_{1340}(459,\cdot)\) \(\chi_{1340}(519,\cdot)\) \(\chi_{1340}(599,\cdot)\) \(\chi_{1340}(739,\cdot)\) \(\chi_{1340}(899,\cdot)\) \(\chi_{1340}(919,\cdot)\) \(\chi_{1340}(979,\cdot)\) \(\chi_{1340}(999,\cdot)\) \(\chi_{1340}(1039,\cdot)\) \(\chi_{1340}(1079,\cdot)\) \(\chi_{1340}(1159,\cdot)\) \(\chi_{1340}(1219,\cdot)\) \(\chi_{1340}(1319,\cdot)\)
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
\((671,537,1141)\) → \((-1,-1,e\left(\frac{41}{66}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 1340 }(79, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{15}{22}\right)\) |
sage: chi.jacobi_sum(n)