from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3024, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,14,0]))
pari: [g,chi] = znchar(Mod(155,3024))
Basic properties
| Modulus: | \(3024\) | |
| Conductor: | \(432\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{432}(155,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3024.he
\(\chi_{3024}(155,\cdot)\) \(\chi_{3024}(491,\cdot)\) \(\chi_{3024}(659,\cdot)\) \(\chi_{3024}(995,\cdot)\) \(\chi_{3024}(1163,\cdot)\) \(\chi_{3024}(1499,\cdot)\) \(\chi_{3024}(1667,\cdot)\) \(\chi_{3024}(2003,\cdot)\) \(\chi_{3024}(2171,\cdot)\) \(\chi_{3024}(2507,\cdot)\) \(\chi_{3024}(2675,\cdot)\) \(\chi_{3024}(3011,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | 36.36.5532004127928253705369187176396364210546696053048780432717505515499814912.1 |
Values on generators
\((1135,757,785,2593)\) → \((-1,i,e\left(\frac{7}{18}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 3024 }(155, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{7}{12}\right)\) |
sage: chi.jacobi_sum(n)