from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3645, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([17,9]))
pari: [g,chi] = znchar(Mod(2834,3645))
Basic properties
| Modulus: | \(3645\) | |
| Conductor: | \(135\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(18\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{135}(14,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3645.n
\(\chi_{3645}(404,\cdot)\) \(\chi_{3645}(809,\cdot)\) \(\chi_{3645}(1619,\cdot)\) \(\chi_{3645}(2024,\cdot)\) \(\chi_{3645}(2834,\cdot)\) \(\chi_{3645}(3239,\cdot)\)
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: | 18.0.5770142004982097067662109375.1 |
Values on generators
\((731,2917)\) → \((e\left(\frac{17}{18}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 3645 }(2834, a) \) | \(-1\) | \(1\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) |
sage: chi.jacobi_sum(n)