from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5520, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,11,0,0,15]))
pari: [g,chi] = znchar(Mod(4711,5520))
Basic properties
| Modulus: | \(5520\) | |
| Conductor: | \(184\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{184}(19,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5520.dr
\(\chi_{5520}(631,\cdot)\) \(\chi_{5520}(871,\cdot)\) \(\chi_{5520}(1111,\cdot)\) \(\chi_{5520}(1351,\cdot)\) \(\chi_{5520}(1831,\cdot)\) \(\chi_{5520}(2311,\cdot)\) \(\chi_{5520}(2551,\cdot)\) \(\chi_{5520}(3271,\cdot)\) \(\chi_{5520}(3511,\cdot)\) \(\chi_{5520}(4711,\cdot)\)
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: | \(\Q(\zeta_{11})\) |
| Fixed field: | 22.22.339058325839400057321133061640411938816.1 |
Values on generators
\((4831,1381,1841,4417,1201)\) → \((-1,-1,1,1,e\left(\frac{15}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 5520 }(4711, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) |
sage: chi.jacobi_sum(n)