from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1344, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([0,9,12,20]))
pari: [g,chi] = znchar(Mod(1097,1344))
Basic properties
| Modulus: | \(1344\) | |
| Conductor: | \(672\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{672}(509,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1344.ct
\(\chi_{1344}(89,\cdot)\) \(\chi_{1344}(185,\cdot)\) \(\chi_{1344}(425,\cdot)\) \(\chi_{1344}(521,\cdot)\) \(\chi_{1344}(761,\cdot)\) \(\chi_{1344}(857,\cdot)\) \(\chi_{1344}(1097,\cdot)\) \(\chi_{1344}(1193,\cdot)\)
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: | \(\Q(\zeta_{24})\) |
| Fixed field: | 24.24.419957608266393538093113781921531638278568932278272.1 |
Values on generators
\((127,1093,449,577)\) → \((1,e\left(\frac{3}{8}\right),-1,e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 1344 }(1097, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{24}\right)\) |
sage: chi.jacobi_sum(n)