from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9576, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([9,0,12,9,16]))
pari: [g,chi] = znchar(Mod(727,9576))
Basic properties
Modulus: | \(9576\) | |
Conductor: | \(4788\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(18\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{4788}(727,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9576.yo
\(\chi_{9576}(727,\cdot)\) \(\chi_{9576}(2239,\cdot)\) \(\chi_{9576}(2911,\cdot)\) \(\chi_{9576}(3247,\cdot)\) \(\chi_{9576}(4927,\cdot)\) \(\chi_{9576}(7951,\cdot)\)
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
\((7183,4789,5321,4105,1009)\) → \((-1,1,e\left(\frac{2}{3}\right),-1,e\left(\frac{8}{9}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 9576 }(727, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{7}{18}\right)\) |
sage: chi.jacobi_sum(n)