from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4830, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,33,22,30]))
pari: [g,chi] = znchar(Mod(3229,4830))
Basic properties
| Modulus: | \(4830\) | |
| Conductor: | \(805\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{805}(9,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4830.di
\(\chi_{4830}(289,\cdot)\) \(\chi_{4830}(499,\cdot)\) \(\chi_{4830}(739,\cdot)\) \(\chi_{4830}(949,\cdot)\) \(\chi_{4830}(1129,\cdot)\) \(\chi_{4830}(1159,\cdot)\) \(\chi_{4830}(1369,\cdot)\) \(\chi_{4830}(1549,\cdot)\) \(\chi_{4830}(1789,\cdot)\) \(\chi_{4830}(2419,\cdot)\) \(\chi_{4830}(2809,\cdot)\) \(\chi_{4830}(3019,\cdot)\) \(\chi_{4830}(3049,\cdot)\) \(\chi_{4830}(3229,\cdot)\) \(\chi_{4830}(3259,\cdot)\) \(\chi_{4830}(3439,\cdot)\) \(\chi_{4830}(3859,\cdot)\) \(\chi_{4830}(3889,\cdot)\) \(\chi_{4830}(4309,\cdot)\) \(\chi_{4830}(4489,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
| Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((3221,967,2761,1891)\) → \((1,-1,e\left(\frac{1}{3}\right),e\left(\frac{5}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 4830 }(3229, a) \) | \(1\) | \(1\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{1}{6}\right)\) |
sage: chi.jacobi_sum(n)