from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4032, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,21,40,32]))
pari: [g,chi] = znchar(Mod(851,4032))
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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4032.ha
\(\chi_{4032}(347,\cdot)\) \(\chi_{4032}(443,\cdot)\) \(\chi_{4032}(851,\cdot)\) \(\chi_{4032}(947,\cdot)\) \(\chi_{4032}(1355,\cdot)\) \(\chi_{4032}(1451,\cdot)\) \(\chi_{4032}(1859,\cdot)\) \(\chi_{4032}(1955,\cdot)\) \(\chi_{4032}(2363,\cdot)\) \(\chi_{4032}(2459,\cdot)\) \(\chi_{4032}(2867,\cdot)\) \(\chi_{4032}(2963,\cdot)\) \(\chi_{4032}(3371,\cdot)\) \(\chi_{4032}(3467,\cdot)\) \(\chi_{4032}(3875,\cdot)\) \(\chi_{4032}(3971,\cdot)\)
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
\((127,3781,1793,577)\) → \((-1,e\left(\frac{7}{16}\right),e\left(\frac{5}{6}\right),e\left(\frac{2}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 4032 }(851, a) \) | \(1\) | \(1\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{13}{48}\right)\) |
sage: chi.jacobi_sum(n)