from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2808, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([9,9,4,9]))
pari: [g,chi] = znchar(Mod(259,2808))
Basic properties
| Modulus: | \(2808\) | |
| Conductor: | \(2808\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(18\) | 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 2808.fi
\(\chi_{2808}(259,\cdot)\) \(\chi_{2808}(571,\cdot)\) \(\chi_{2808}(1195,\cdot)\) \(\chi_{2808}(1507,\cdot)\) \(\chi_{2808}(2131,\cdot)\) \(\chi_{2808}(2443,\cdot)\)
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: | Number field defined by a degree 18 polynomial |
Values on generators
\((703,1405,2081,1081)\) → \((-1,-1,e\left(\frac{2}{9}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 2808 }(259, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)