from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6776, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,11,42]))
pari: [g,chi] = znchar(Mod(507,6776))
Basic properties
| Modulus: | \(6776\) | |
| Conductor: | \(6776\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6776.dp
\(\chi_{6776}(507,\cdot)\) \(\chi_{6776}(859,\cdot)\) \(\chi_{6776}(1123,\cdot)\) \(\chi_{6776}(1475,\cdot)\) \(\chi_{6776}(1739,\cdot)\) \(\chi_{6776}(2091,\cdot)\) \(\chi_{6776}(2355,\cdot)\) \(\chi_{6776}(2707,\cdot)\) \(\chi_{6776}(2971,\cdot)\) \(\chi_{6776}(3323,\cdot)\) \(\chi_{6776}(3587,\cdot)\) \(\chi_{6776}(3939,\cdot)\) \(\chi_{6776}(4203,\cdot)\) \(\chi_{6776}(4555,\cdot)\) \(\chi_{6776}(4819,\cdot)\) \(\chi_{6776}(5171,\cdot)\) \(\chi_{6776}(5435,\cdot)\) \(\chi_{6776}(5787,\cdot)\) \(\chi_{6776}(6403,\cdot)\) \(\chi_{6776}(6667,\cdot)\)
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
\((1695,3389,969,3753)\) → \((-1,-1,e\left(\frac{1}{6}\right),e\left(\frac{7}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 6776 }(507, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)