from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6776, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,0,11,48]))
pari: [g,chi] = znchar(Mod(199,6776))
Basic properties
| Modulus: | \(6776\) | |
| Conductor: | \(3388\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{3388}(199,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6776.dg
\(\chi_{6776}(199,\cdot)\) \(\chi_{6776}(551,\cdot)\) \(\chi_{6776}(815,\cdot)\) \(\chi_{6776}(1167,\cdot)\) \(\chi_{6776}(1431,\cdot)\) \(\chi_{6776}(1783,\cdot)\) \(\chi_{6776}(2047,\cdot)\) \(\chi_{6776}(2399,\cdot)\) \(\chi_{6776}(3015,\cdot)\) \(\chi_{6776}(3279,\cdot)\) \(\chi_{6776}(3895,\cdot)\) \(\chi_{6776}(4247,\cdot)\) \(\chi_{6776}(4511,\cdot)\) \(\chi_{6776}(4863,\cdot)\) \(\chi_{6776}(5127,\cdot)\) \(\chi_{6776}(5479,\cdot)\) \(\chi_{6776}(5743,\cdot)\) \(\chi_{6776}(6095,\cdot)\) \(\chi_{6776}(6359,\cdot)\) \(\chi_{6776}(6711,\cdot)\)
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
\((1695,3389,969,3753)\) → \((-1,1,e\left(\frac{1}{6}\right),e\left(\frac{8}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 6776 }(199, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)