from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6776, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,0,11,4]))
pari: [g,chi] = znchar(Mod(111,6776))
Basic properties
| Modulus: | \(6776\) | |
| Conductor: | \(3388\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{3388}(111,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6776.cm
\(\chi_{6776}(111,\cdot)\) \(\chi_{6776}(1343,\cdot)\) \(\chi_{6776}(1959,\cdot)\) \(\chi_{6776}(2575,\cdot)\) \(\chi_{6776}(3191,\cdot)\) \(\chi_{6776}(3807,\cdot)\) \(\chi_{6776}(4423,\cdot)\) \(\chi_{6776}(5039,\cdot)\) \(\chi_{6776}(5655,\cdot)\) \(\chi_{6776}(6271,\cdot)\)
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: | \(\Q(\zeta_{11})\) |
| Fixed field: | Number field defined by a degree 22 polynomial |
Values on generators
\((1695,3389,969,3753)\) → \((-1,1,-1,e\left(\frac{2}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 6776 }(111, a) \) | \(1\) | \(1\) | \(1\) | \(e\left(\frac{21}{22}\right)\) | \(1\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)