from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6776, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,0,44,57]))
pari: [g,chi] = znchar(Mod(263,6776))
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 \(\chi_{3388}(263,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6776.dr
\(\chi_{6776}(263,\cdot)\) \(\chi_{6776}(527,\cdot)\) \(\chi_{6776}(879,\cdot)\) \(\chi_{6776}(1143,\cdot)\) \(\chi_{6776}(1495,\cdot)\) \(\chi_{6776}(1759,\cdot)\) \(\chi_{6776}(2111,\cdot)\) \(\chi_{6776}(2375,\cdot)\) \(\chi_{6776}(2727,\cdot)\) \(\chi_{6776}(2991,\cdot)\) \(\chi_{6776}(3343,\cdot)\) \(\chi_{6776}(3607,\cdot)\) \(\chi_{6776}(3959,\cdot)\) \(\chi_{6776}(4223,\cdot)\) \(\chi_{6776}(4575,\cdot)\) \(\chi_{6776}(5191,\cdot)\) \(\chi_{6776}(5455,\cdot)\) \(\chi_{6776}(6071,\cdot)\) \(\chi_{6776}(6423,\cdot)\) \(\chi_{6776}(6687,\cdot)\)
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
\((1695,3389,969,3753)\) → \((-1,1,e\left(\frac{2}{3}\right),e\left(\frac{19}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
\( \chi_{ 6776 }(263, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)