from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6048, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,8,18]))
pari: [g,chi] = znchar(Mod(5767,6048))
Basic properties
| Modulus: | \(6048\) | |
| Conductor: | \(3024\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{3024}(475,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6048.in
\(\chi_{6048}(391,\cdot)\) \(\chi_{6048}(727,\cdot)\) \(\chi_{6048}(1399,\cdot)\) \(\chi_{6048}(1735,\cdot)\) \(\chi_{6048}(2407,\cdot)\) \(\chi_{6048}(2743,\cdot)\) \(\chi_{6048}(3415,\cdot)\) \(\chi_{6048}(3751,\cdot)\) \(\chi_{6048}(4423,\cdot)\) \(\chi_{6048}(4759,\cdot)\) \(\chi_{6048}(5431,\cdot)\) \(\chi_{6048}(5767,\cdot)\)
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: | Number field defined by a degree 36 polynomial |
Values on generators
\((4159,3781,3809,2593)\) → \((-1,i,e\left(\frac{2}{9}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 6048 }(5767, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{7}{12}\right)\) |
sage: chi.jacobi_sum(n)