from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2898, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,55,15]))
pari: [g,chi] = znchar(Mod(89,2898))
Basic properties
| Modulus: | \(2898\) | |
| Conductor: | \(483\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{483}(89,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2898.co
\(\chi_{2898}(17,\cdot)\) \(\chi_{2898}(89,\cdot)\) \(\chi_{2898}(143,\cdot)\) \(\chi_{2898}(341,\cdot)\) \(\chi_{2898}(467,\cdot)\) \(\chi_{2898}(521,\cdot)\) \(\chi_{2898}(773,\cdot)\) \(\chi_{2898}(845,\cdot)\) \(\chi_{2898}(971,\cdot)\) \(\chi_{2898}(1349,\cdot)\) \(\chi_{2898}(1529,\cdot)\) \(\chi_{2898}(1601,\cdot)\) \(\chi_{2898}(1781,\cdot)\) \(\chi_{2898}(1907,\cdot)\) \(\chi_{2898}(2159,\cdot)\) \(\chi_{2898}(2357,\cdot)\) \(\chi_{2898}(2411,\cdot)\) \(\chi_{2898}(2537,\cdot)\) \(\chi_{2898}(2609,\cdot)\) \(\chi_{2898}(2735,\cdot)\)
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
\((1289,829,1891)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{5}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 2898 }(89, a) \) | \(-1\) | \(1\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{8}{11}\right)\) |
sage: chi.jacobi_sum(n)