from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6048, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,2,12]))
pari: [g,chi] = znchar(Mod(1703,6048))
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 \(\chi_{3024}(2459,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6048.iq
\(\chi_{6048}(599,\cdot)\) \(\chi_{6048}(695,\cdot)\) \(\chi_{6048}(1607,\cdot)\) \(\chi_{6048}(1703,\cdot)\) \(\chi_{6048}(2615,\cdot)\) \(\chi_{6048}(2711,\cdot)\) \(\chi_{6048}(3623,\cdot)\) \(\chi_{6048}(3719,\cdot)\) \(\chi_{6048}(4631,\cdot)\) \(\chi_{6048}(4727,\cdot)\) \(\chi_{6048}(5639,\cdot)\) \(\chi_{6048}(5735,\cdot)\)
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
\((4159,3781,3809,2593)\) → \((-1,i,e\left(\frac{1}{18}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 6048 }(1703, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(i\) |
sage: chi.jacobi_sum(n)