from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1520, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([0,9,9,5]))
pari: [g,chi] = znchar(Mod(89,1520))
Basic properties
| Modulus: | \(1520\) | |
| Conductor: | \(760\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(18\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{760}(469,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1520.dq
\(\chi_{1520}(89,\cdot)\) \(\chi_{1520}(249,\cdot)\) \(\chi_{1520}(409,\cdot)\) \(\chi_{1520}(489,\cdot)\) \(\chi_{1520}(649,\cdot)\) \(\chi_{1520}(889,\cdot)\)
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: | \(\Q(\zeta_{9})\) |
| Fixed field: | 18.0.1436650532447139184230793216000000000.1 |
Values on generators
\((191,1141,1217,401)\) → \((1,-1,-1,e\left(\frac{5}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 1520 }(89, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{9}\right)\) |
sage: chi.jacobi_sum(n)