from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1053, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([4,6]))
pari: [g,chi] = znchar(Mod(289,1053))
Basic properties
| Modulus: | \(1053\) | |
| Conductor: | \(351\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(9\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{351}(16,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1053.x
\(\chi_{1053}(289,\cdot)\) \(\chi_{1053}(334,\cdot)\) \(\chi_{1053}(640,\cdot)\) \(\chi_{1053}(685,\cdot)\) \(\chi_{1053}(991,\cdot)\) \(\chi_{1053}(1036,\cdot)\)
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: | 9.9.151470380950257681.1 |
Values on generators
\((326,730)\) → \((e\left(\frac{2}{9}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 1053 }(289, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)