from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4032, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,21,8,24]))
pari: [g,chi] = znchar(Mod(83,4032))
Basic properties
| Modulus: | \(4032\) | |
| Conductor: | \(4032\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4032.hl
\(\chi_{4032}(83,\cdot)\) \(\chi_{4032}(419,\cdot)\) \(\chi_{4032}(587,\cdot)\) \(\chi_{4032}(923,\cdot)\) \(\chi_{4032}(1091,\cdot)\) \(\chi_{4032}(1427,\cdot)\) \(\chi_{4032}(1595,\cdot)\) \(\chi_{4032}(1931,\cdot)\) \(\chi_{4032}(2099,\cdot)\) \(\chi_{4032}(2435,\cdot)\) \(\chi_{4032}(2603,\cdot)\) \(\chi_{4032}(2939,\cdot)\) \(\chi_{4032}(3107,\cdot)\) \(\chi_{4032}(3443,\cdot)\) \(\chi_{4032}(3611,\cdot)\) \(\chi_{4032}(3947,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
| Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((127,3781,1793,577)\) → \((-1,e\left(\frac{7}{16}\right),e\left(\frac{1}{6}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 4032 }(83, a) \) | \(-1\) | \(1\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(i\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{15}{16}\right)\) |
sage: chi.jacobi_sum(n)