from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4830, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,0,55,51]))
pari: [g,chi] = znchar(Mod(61,4830))
Basic properties
| Modulus: | \(4830\) | |
| Conductor: | \(161\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{161}(61,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4830.db
\(\chi_{4830}(61,\cdot)\) \(\chi_{4830}(241,\cdot)\) \(\chi_{4830}(451,\cdot)\) \(\chi_{4830}(481,\cdot)\) \(\chi_{4830}(661,\cdot)\) \(\chi_{4830}(871,\cdot)\) \(\chi_{4830}(1111,\cdot)\) \(\chi_{4830}(1321,\cdot)\) \(\chi_{4830}(1951,\cdot)\) \(\chi_{4830}(2131,\cdot)\) \(\chi_{4830}(2551,\cdot)\) \(\chi_{4830}(2581,\cdot)\) \(\chi_{4830}(3001,\cdot)\) \(\chi_{4830}(3181,\cdot)\) \(\chi_{4830}(3211,\cdot)\) \(\chi_{4830}(3391,\cdot)\) \(\chi_{4830}(3421,\cdot)\) \(\chi_{4830}(3631,\cdot)\) \(\chi_{4830}(4021,\cdot)\) \(\chi_{4830}(4651,\cdot)\)
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
\((3221,967,2761,1891)\) → \((1,1,e\left(\frac{5}{6}\right),e\left(\frac{17}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 4830 }(61, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{1}{6}\right)\) |
sage: chi.jacobi_sum(n)