from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5415, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([9,9,17]))
pari: [g,chi] = znchar(Mod(4304,5415))
Basic properties
| Modulus: | \(5415\) | |
| Conductor: | \(285\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(18\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{285}(29,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5415.bf
\(\chi_{5415}(299,\cdot)\) \(\chi_{5415}(1199,\cdot)\) \(\chi_{5415}(2654,\cdot)\) \(\chi_{5415}(2789,\cdot)\) \(\chi_{5415}(2834,\cdot)\) \(\chi_{5415}(4304,\cdot)\)
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
\((3611,2167,5056)\) → \((-1,-1,e\left(\frac{17}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(22\) |
| \( \chi_{ 5415 }(4304, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) |
sage: chi.jacobi_sum(n)