from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9072, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([0,9,14,12]))
pari: [g,chi] = znchar(Mod(361,9072))
Basic properties
| Modulus: | \(9072\) | |
| Conductor: | \(1512\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(18\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1512}(1453,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9072.fi
\(\chi_{9072}(361,\cdot)\) \(\chi_{9072}(1801,\cdot)\) \(\chi_{9072}(3385,\cdot)\) \(\chi_{9072}(4825,\cdot)\) \(\chi_{9072}(6409,\cdot)\) \(\chi_{9072}(7849,\cdot)\)
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
\((1135,6805,3809,2593)\) → \((1,-1,e\left(\frac{7}{9}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 9072 }(361, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)