from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4032, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,45,16,24]))
pari: [g,chi] = znchar(Mod(13,4032))
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.gu
\(\chi_{4032}(13,\cdot)\) \(\chi_{4032}(349,\cdot)\) \(\chi_{4032}(517,\cdot)\) \(\chi_{4032}(853,\cdot)\) \(\chi_{4032}(1021,\cdot)\) \(\chi_{4032}(1357,\cdot)\) \(\chi_{4032}(1525,\cdot)\) \(\chi_{4032}(1861,\cdot)\) \(\chi_{4032}(2029,\cdot)\) \(\chi_{4032}(2365,\cdot)\) \(\chi_{4032}(2533,\cdot)\) \(\chi_{4032}(2869,\cdot)\) \(\chi_{4032}(3037,\cdot)\) \(\chi_{4032}(3373,\cdot)\) \(\chi_{4032}(3541,\cdot)\) \(\chi_{4032}(3877,\cdot)\)
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
\((127,3781,1793,577)\) → \((1,e\left(\frac{15}{16}\right),e\left(\frac{1}{3}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 4032 }(13, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(-i\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{16}\right)\) |
sage: chi.jacobi_sum(n)