from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1080, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([9,9,5,9]))
pari: [g,chi] = znchar(Mod(59,1080))
Basic properties
Modulus: | \(1080\) | |
Conductor: | \(1080\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1080.bx
\(\chi_{1080}(59,\cdot)\) \(\chi_{1080}(299,\cdot)\) \(\chi_{1080}(419,\cdot)\) \(\chi_{1080}(659,\cdot)\) \(\chi_{1080}(779,\cdot)\) \(\chi_{1080}(1019,\cdot)\)
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: | 18.18.774455350146061749097070592000000000.1 |
Values on generators
\((271,541,1001,217)\) → \((-1,-1,e\left(\frac{5}{18}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 1080 }(59, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{13}{18}\right)\) |
sage: chi.jacobi_sum(n)