from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1323, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([7,38]))
pari: [g,chi] = znchar(Mod(170,1323))
Basic properties
| Modulus: | \(1323\) | |
| Conductor: | \(441\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{441}(317,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1323.bu
\(\chi_{1323}(170,\cdot)\) \(\chi_{1323}(179,\cdot)\) \(\chi_{1323}(359,\cdot)\) \(\chi_{1323}(368,\cdot)\) \(\chi_{1323}(548,\cdot)\) \(\chi_{1323}(737,\cdot)\) \(\chi_{1323}(746,\cdot)\) \(\chi_{1323}(926,\cdot)\) \(\chi_{1323}(935,\cdot)\) \(\chi_{1323}(1115,\cdot)\) \(\chi_{1323}(1124,\cdot)\) \(\chi_{1323}(1313,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | Number field defined by a degree 42 polynomial |
Values on generators
\((785,1081)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{19}{21}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1323 }(170, a) \) | \(-1\) | \(1\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)