from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1120, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([0,9,18,4]))
pari: [g,chi] = znchar(Mod(1053,1120))
Basic properties
| Modulus: | \(1120\) | |
| Conductor: | \(1120\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1120.ec
\(\chi_{1120}(117,\cdot)\) \(\chi_{1120}(173,\cdot)\) \(\chi_{1120}(437,\cdot)\) \(\chi_{1120}(493,\cdot)\) \(\chi_{1120}(677,\cdot)\) \(\chi_{1120}(733,\cdot)\) \(\chi_{1120}(997,\cdot)\) \(\chi_{1120}(1053,\cdot)\)
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: | 24.24.3014466591648416901112203274753113325568000000000000000000.1 |
Values on generators
\((351,421,897,801)\) → \((1,e\left(\frac{3}{8}\right),-i,e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 1120 }(1053, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{6}\right)\) |
sage: chi.jacobi_sum(n)