from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4032, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,39,40,0]))
pari: [g,chi] = znchar(Mod(491,4032))
Basic properties
Modulus: | \(4032\) | |
Conductor: | \(576\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{576}(491,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4032.hb
\(\chi_{4032}(155,\cdot)\) \(\chi_{4032}(491,\cdot)\) \(\chi_{4032}(659,\cdot)\) \(\chi_{4032}(995,\cdot)\) \(\chi_{4032}(1163,\cdot)\) \(\chi_{4032}(1499,\cdot)\) \(\chi_{4032}(1667,\cdot)\) \(\chi_{4032}(2003,\cdot)\) \(\chi_{4032}(2171,\cdot)\) \(\chi_{4032}(2507,\cdot)\) \(\chi_{4032}(2675,\cdot)\) \(\chi_{4032}(3011,\cdot)\) \(\chi_{4032}(3179,\cdot)\) \(\chi_{4032}(3515,\cdot)\) \(\chi_{4032}(3683,\cdot)\) \(\chi_{4032}(4019,\cdot)\)
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
\((127,3781,1793,577)\) → \((-1,e\left(\frac{13}{16}\right),e\left(\frac{5}{6}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 4032 }(491, a) \) | \(1\) | \(1\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(i\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{16}\right)\) |
sage: chi.jacobi_sum(n)