from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2420, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,11,12]))
pari: [g,chi] = znchar(Mod(1299,2420))
Basic properties
| Modulus: | \(2420\) | |
| Conductor: | \(2420\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2420.bd
\(\chi_{2420}(199,\cdot)\) \(\chi_{2420}(419,\cdot)\) \(\chi_{2420}(639,\cdot)\) \(\chi_{2420}(859,\cdot)\) \(\chi_{2420}(1079,\cdot)\) \(\chi_{2420}(1299,\cdot)\) \(\chi_{2420}(1519,\cdot)\) \(\chi_{2420}(1739,\cdot)\) \(\chi_{2420}(1959,\cdot)\) \(\chi_{2420}(2399,\cdot)\)
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: | Number field defined by a degree 22 polynomial |
Values on generators
\((1211,1937,2301)\) → \((-1,-1,e\left(\frac{6}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 2420 }(1299, a) \) | \(-1\) | \(1\) | \(1\) | \(e\left(\frac{9}{11}\right)\) | \(1\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(1\) | \(e\left(\frac{3}{11}\right)\) |
sage: chi.jacobi_sum(n)