from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1323, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,41]))
pari: [g,chi] = znchar(Mod(1160,1323))
Basic properties
| Modulus: | \(1323\) | |
| Conductor: | \(147\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{147}(131,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1323.bs
\(\chi_{1323}(26,\cdot)\) \(\chi_{1323}(269,\cdot)\) \(\chi_{1323}(404,\cdot)\) \(\chi_{1323}(458,\cdot)\) \(\chi_{1323}(593,\cdot)\) \(\chi_{1323}(647,\cdot)\) \(\chi_{1323}(782,\cdot)\) \(\chi_{1323}(836,\cdot)\) \(\chi_{1323}(971,\cdot)\) \(\chi_{1323}(1025,\cdot)\) \(\chi_{1323}(1160,\cdot)\) \(\chi_{1323}(1214,\cdot)\)
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: | \(\Q(\zeta_{147})^+\) |
Values on generators
\((785,1081)\) → \((-1,e\left(\frac{41}{42}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1323 }(1160, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{1}{6}\right)\) |
sage: chi.jacobi_sum(n)