from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1340, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,33,1]))
pari: [g,chi] = znchar(Mod(69,1340))
Basic properties
| Modulus: | \(1340\) | |
| Conductor: | \(335\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{335}(69,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1340.bm
\(\chi_{1340}(69,\cdot)\) \(\chi_{1340}(229,\cdot)\) \(\chi_{1340}(249,\cdot)\) \(\chi_{1340}(309,\cdot)\) \(\chi_{1340}(329,\cdot)\) \(\chi_{1340}(369,\cdot)\) \(\chi_{1340}(409,\cdot)\) \(\chi_{1340}(489,\cdot)\) \(\chi_{1340}(549,\cdot)\) \(\chi_{1340}(649,\cdot)\) \(\chi_{1340}(749,\cdot)\) \(\chi_{1340}(769,\cdot)\) \(\chi_{1340}(889,\cdot)\) \(\chi_{1340}(949,\cdot)\) \(\chi_{1340}(969,\cdot)\) \(\chi_{1340}(989,\cdot)\) \(\chi_{1340}(1049,\cdot)\) \(\chi_{1340}(1129,\cdot)\) \(\chi_{1340}(1189,\cdot)\) \(\chi_{1340}(1269,\cdot)\)
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
\((671,537,1141)\) → \((1,-1,e\left(\frac{1}{66}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 1340 }(69, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{3}{11}\right)\) |
sage: chi.jacobi_sum(n)