from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1265, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([11,22,26]))
pari: [g,chi] = znchar(Mod(527,1265))
Basic properties
| Modulus: | \(1265\) | |
| Conductor: | \(1265\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1265.bi
\(\chi_{1265}(43,\cdot)\) \(\chi_{1265}(153,\cdot)\) \(\chi_{1265}(263,\cdot)\) \(\chi_{1265}(318,\cdot)\) \(\chi_{1265}(362,\cdot)\) \(\chi_{1265}(373,\cdot)\) \(\chi_{1265}(428,\cdot)\) \(\chi_{1265}(527,\cdot)\) \(\chi_{1265}(582,\cdot)\) \(\chi_{1265}(747,\cdot)\) \(\chi_{1265}(802,\cdot)\) \(\chi_{1265}(868,\cdot)\) \(\chi_{1265}(912,\cdot)\) \(\chi_{1265}(1022,\cdot)\) \(\chi_{1265}(1033,\cdot)\) \(\chi_{1265}(1077,\cdot)\) \(\chi_{1265}(1088,\cdot)\) \(\chi_{1265}(1132,\cdot)\) \(\chi_{1265}(1187,\cdot)\) \(\chi_{1265}(1253,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | Number field defined by a degree 44 polynomial |
Values on generators
\((507,1036,166)\) → \((i,-1,e\left(\frac{13}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 1265 }(527, a) \) | \(-1\) | \(1\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{10}{11}\right)\) |
sage: chi.jacobi_sum(n)