from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1449, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,11,9]))
pari: [g,chi] = znchar(Mod(10,1449))
Basic properties
| Modulus: | \(1449\) | |
| 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 \(\chi_{161}(10,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1449.cv
\(\chi_{1449}(10,\cdot)\) \(\chi_{1449}(19,\cdot)\) \(\chi_{1449}(136,\cdot)\) \(\chi_{1449}(145,\cdot)\) \(\chi_{1449}(199,\cdot)\) \(\chi_{1449}(388,\cdot)\) \(\chi_{1449}(451,\cdot)\) \(\chi_{1449}(523,\cdot)\) \(\chi_{1449}(586,\cdot)\) \(\chi_{1449}(640,\cdot)\) \(\chi_{1449}(649,\cdot)\) \(\chi_{1449}(766,\cdot)\) \(\chi_{1449}(838,\cdot)\) \(\chi_{1449}(964,\cdot)\) \(\chi_{1449}(1027,\cdot)\) \(\chi_{1449}(1144,\cdot)\) \(\chi_{1449}(1207,\cdot)\) \(\chi_{1449}(1216,\cdot)\) \(\chi_{1449}(1270,\cdot)\) \(\chi_{1449}(1279,\cdot)\)
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
\((1289,829,442)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{3}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1449 }(10, a) \) | \(1\) | \(1\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) |
sage: chi.jacobi_sum(n)