from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4284, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,32,40,9]))
pari: [g,chi] = znchar(Mod(61,4284))
Basic properties
| Modulus: | \(4284\) | |
| Conductor: | \(1071\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1071}(61,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4284.hm
\(\chi_{4284}(61,\cdot)\) \(\chi_{4284}(313,\cdot)\) \(\chi_{4284}(913,\cdot)\) \(\chi_{4284}(1321,\cdot)\) \(\chi_{4284}(1417,\cdot)\) \(\chi_{4284}(1669,\cdot)\) \(\chi_{4284}(1825,\cdot)\) \(\chi_{4284}(2077,\cdot)\) \(\chi_{4284}(2173,\cdot)\) \(\chi_{4284}(2425,\cdot)\) \(\chi_{4284}(2581,\cdot)\) \(\chi_{4284}(2833,\cdot)\) \(\chi_{4284}(2929,\cdot)\) \(\chi_{4284}(3337,\cdot)\) \(\chi_{4284}(3937,\cdot)\) \(\chi_{4284}(4189,\cdot)\)
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
\((2143,3809,1837,1261)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{5}{6}\right),e\left(\frac{3}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 4284 }(61, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{43}{48}\right)\) |
sage: chi.jacobi_sum(n)