from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1323, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,41]))
pari: [g,chi] = znchar(Mod(82,1323))
Basic properties
| Modulus: | \(1323\) | |
| Conductor: | \(49\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{49}(33,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1323.bv
\(\chi_{1323}(82,\cdot)\) \(\chi_{1323}(136,\cdot)\) \(\chi_{1323}(271,\cdot)\) \(\chi_{1323}(514,\cdot)\) \(\chi_{1323}(649,\cdot)\) \(\chi_{1323}(703,\cdot)\) \(\chi_{1323}(838,\cdot)\) \(\chi_{1323}(892,\cdot)\) \(\chi_{1323}(1027,\cdot)\) \(\chi_{1323}(1081,\cdot)\) \(\chi_{1323}(1216,\cdot)\) \(\chi_{1323}(1270,\cdot)\)
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
\((785,1081)\) → \((1,e\left(\frac{41}{42}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1323 }(82, a) \) | \(-1\) | \(1\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{1}{6}\right)\) |
sage: chi.jacobi_sum(n)