from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4032, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,15,8,8]))
pari: [g,chi] = znchar(Mod(3125,4032))
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.hp
\(\chi_{4032}(5,\cdot)\) \(\chi_{4032}(101,\cdot)\) \(\chi_{4032}(509,\cdot)\) \(\chi_{4032}(605,\cdot)\) \(\chi_{4032}(1013,\cdot)\) \(\chi_{4032}(1109,\cdot)\) \(\chi_{4032}(1517,\cdot)\) \(\chi_{4032}(1613,\cdot)\) \(\chi_{4032}(2021,\cdot)\) \(\chi_{4032}(2117,\cdot)\) \(\chi_{4032}(2525,\cdot)\) \(\chi_{4032}(2621,\cdot)\) \(\chi_{4032}(3029,\cdot)\) \(\chi_{4032}(3125,\cdot)\) \(\chi_{4032}(3533,\cdot)\) \(\chi_{4032}(3629,\cdot)\)
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
\((127,3781,1793,577)\) → \((1,e\left(\frac{5}{16}\right),e\left(\frac{1}{6}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 4032 }(3125, a) \) | \(1\) | \(1\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(1\) | \(e\left(\frac{7}{48}\right)\) |
sage: chi.jacobi_sum(n)