from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1428, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,0,40,9]))
pari: [g,chi] = znchar(Mod(61,1428))
Basic properties
| Modulus: | \(1428\) | |
| Conductor: | \(119\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{119}(61,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1428.cz
\(\chi_{1428}(61,\cdot)\) \(\chi_{1428}(73,\cdot)\) \(\chi_{1428}(241,\cdot)\) \(\chi_{1428}(313,\cdot)\) \(\chi_{1428}(397,\cdot)\) \(\chi_{1428}(481,\cdot)\) \(\chi_{1428}(649,\cdot)\) \(\chi_{1428}(745,\cdot)\) \(\chi_{1428}(913,\cdot)\) \(\chi_{1428}(997,\cdot)\) \(\chi_{1428}(1081,\cdot)\) \(\chi_{1428}(1153,\cdot)\) \(\chi_{1428}(1321,\cdot)\) \(\chi_{1428}(1333,\cdot)\) \(\chi_{1428}(1405,\cdot)\) \(\chi_{1428}(1417,\cdot)\)
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
\((715,953,409,1261)\) → \((1,1,e\left(\frac{5}{6}\right),e\left(\frac{3}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 1428 }(61, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(i\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{9}{16}\right)\) |
sage: chi.jacobi_sum(n)