from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3330, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([6,0,2]))
pari: [g,chi] = znchar(Mod(571,3330))
Basic properties
Modulus: | \(3330\) | |
Conductor: | \(333\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(9\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{333}(238,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3330.ca
\(\chi_{3330}(571,\cdot)\) \(\chi_{3330}(1381,\cdot)\) \(\chi_{3330}(2401,\cdot)\) \(\chi_{3330}(2671,\cdot)\) \(\chi_{3330}(2821,\cdot)\) \(\chi_{3330}(3031,\cdot)\)
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: | 9.9.1866675593471230161.1 |
Values on generators
\((371,667,631)\) → \((e\left(\frac{1}{3}\right),1,e\left(\frac{1}{9}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(41\) | \(43\) |
\( \chi_{ 3330 }(571, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) |
sage: chi.jacobi_sum(n)