from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8280, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,0,22,0,18]))
pari: [g,chi] = znchar(Mod(31,8280))
Basic properties
| Modulus: | \(8280\) | |
| Conductor: | \(828\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{828}(31,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8280.fu
\(\chi_{8280}(31,\cdot)\) \(\chi_{8280}(151,\cdot)\) \(\chi_{8280}(1231,\cdot)\) \(\chi_{8280}(1591,\cdot)\) \(\chi_{8280}(2191,\cdot)\) \(\chi_{8280}(2671,\cdot)\) \(\chi_{8280}(2911,\cdot)\) \(\chi_{8280}(3031,\cdot)\) \(\chi_{8280}(3751,\cdot)\) \(\chi_{8280}(3991,\cdot)\) \(\chi_{8280}(4351,\cdot)\) \(\chi_{8280}(4471,\cdot)\) \(\chi_{8280}(5191,\cdot)\) \(\chi_{8280}(5431,\cdot)\) \(\chi_{8280}(5551,\cdot)\) \(\chi_{8280}(5791,\cdot)\) \(\chi_{8280}(6511,\cdot)\) \(\chi_{8280}(7231,\cdot)\) \(\chi_{8280}(7711,\cdot)\) \(\chi_{8280}(7951,\cdot)\)
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
\((2071,4141,4601,1657,3961)\) → \((-1,1,e\left(\frac{1}{3}\right),1,e\left(\frac{3}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 8280 }(31, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{13}{66}\right)\) |
sage: chi.jacobi_sum(n)