from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4032, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,21,16,16]))
pari: [g,chi] = znchar(Mod(877,4032))
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.go
\(\chi_{4032}(277,\cdot)\) \(\chi_{4032}(373,\cdot)\) \(\chi_{4032}(781,\cdot)\) \(\chi_{4032}(877,\cdot)\) \(\chi_{4032}(1285,\cdot)\) \(\chi_{4032}(1381,\cdot)\) \(\chi_{4032}(1789,\cdot)\) \(\chi_{4032}(1885,\cdot)\) \(\chi_{4032}(2293,\cdot)\) \(\chi_{4032}(2389,\cdot)\) \(\chi_{4032}(2797,\cdot)\) \(\chi_{4032}(2893,\cdot)\) \(\chi_{4032}(3301,\cdot)\) \(\chi_{4032}(3397,\cdot)\) \(\chi_{4032}(3805,\cdot)\) \(\chi_{4032}(3901,\cdot)\)
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
\((127,3781,1793,577)\) → \((1,e\left(\frac{7}{16}\right),e\left(\frac{1}{3}\right),e\left(\frac{1}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 4032 }(877, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(-1\) | \(e\left(\frac{29}{48}\right)\) |
sage: chi.jacobi_sum(n)