from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9576, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([0,0,12,0,10]))
pari: [g,chi] = znchar(Mod(169,9576))
Basic properties
| Modulus: | \(9576\) | |
| Conductor: | \(171\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(9\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{171}(169,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9576.rj
\(\chi_{9576}(169,\cdot)\) \(\chi_{9576}(1849,\cdot)\) \(\chi_{9576}(4873,\cdot)\) \(\chi_{9576}(7225,\cdot)\) \(\chi_{9576}(8737,\cdot)\) \(\chi_{9576}(9409,\cdot)\)
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: | 9.9.9025761726072081.2 |
Values on generators
\((7183,4789,5321,4105,1009)\) → \((1,1,e\left(\frac{2}{3}\right),1,e\left(\frac{5}{9}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 9576 }(169, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{5}{9}\right)\) |
sage: chi.jacobi_sum(n)