from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7728, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,0,22,21]))
pari: [g,chi] = znchar(Mod(247,7728))
Basic properties
| Modulus: | \(7728\) | |
| Conductor: | \(1288\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1288}(891,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7728.gj
\(\chi_{7728}(247,\cdot)\) \(\chi_{7728}(1831,\cdot)\) \(\chi_{7728}(2167,\cdot)\) \(\chi_{7728}(2503,\cdot)\) \(\chi_{7728}(2839,\cdot)\) \(\chi_{7728}(2935,\cdot)\) \(\chi_{7728}(3271,\cdot)\) \(\chi_{7728}(3511,\cdot)\) \(\chi_{7728}(3607,\cdot)\) \(\chi_{7728}(3943,\cdot)\) \(\chi_{7728}(4183,\cdot)\) \(\chi_{7728}(4519,\cdot)\) \(\chi_{7728}(4615,\cdot)\) \(\chi_{7728}(5287,\cdot)\) \(\chi_{7728}(5527,\cdot)\) \(\chi_{7728}(5623,\cdot)\) \(\chi_{7728}(5863,\cdot)\) \(\chi_{7728}(6631,\cdot)\) \(\chi_{7728}(6871,\cdot)\) \(\chi_{7728}(6967,\cdot)\)
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
\((4831,5797,5153,6625,6721)\) → \((-1,-1,1,e\left(\frac{1}{3}\right),e\left(\frac{7}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 7728 }(247, a) \) | \(1\) | \(1\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{9}{11}\right)\) |
sage: chi.jacobi_sum(n)