from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8550, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([9,0,5]))
pari: [g,chi] = znchar(Mod(3851,8550))
Basic properties
Modulus: | \(8550\) | |
Conductor: | \(57\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(18\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{57}(32,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8550.df
\(\chi_{8550}(3851,\cdot)\) \(\chi_{8550}(5201,\cdot)\) \(\chi_{8550}(6101,\cdot)\) \(\chi_{8550}(6551,\cdot)\) \(\chi_{8550}(7451,\cdot)\) \(\chi_{8550}(8351,\cdot)\)
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: | \(\Q(\zeta_{57})^+\) |
Values on generators
\((1901,1027,1351)\) → \((-1,1,e\left(\frac{5}{18}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 8550 }(3851, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-1\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) |
sage: chi.jacobi_sum(n)