from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2835, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([1,0,12]))
pari: [g,chi] = znchar(Mod(1061,2835))
Basic properties
| Modulus: | \(2835\) | |
| Conductor: | \(189\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(18\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{189}(137,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2835.cv
\(\chi_{2835}(116,\cdot)\) \(\chi_{2835}(611,\cdot)\) \(\chi_{2835}(1061,\cdot)\) \(\chi_{2835}(1556,\cdot)\) \(\chi_{2835}(2006,\cdot)\) \(\chi_{2835}(2501,\cdot)\)
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
\((1541,1702,2026)\) → \((e\left(\frac{1}{18}\right),1,e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 2835 }(1061, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{17}{18}\right)\) |
sage: chi.jacobi_sum(n)