from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6762, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,55,27]))
pari: [g,chi] = znchar(Mod(5899,6762))
Basic properties
Modulus: | \(6762\) | |
Conductor: | \(161\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{161}(103,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6762.bu
\(\chi_{6762}(19,\cdot)\) \(\chi_{6762}(313,\cdot)\) \(\chi_{6762}(619,\cdot)\) \(\chi_{6762}(1207,\cdot)\) \(\chi_{6762}(1489,\cdot)\) \(\chi_{6762}(2077,\cdot)\) \(\chi_{6762}(2089,\cdot)\) \(\chi_{6762}(2383,\cdot)\) \(\chi_{6762}(2665,\cdot)\) \(\chi_{6762}(2959,\cdot)\) \(\chi_{6762}(3253,\cdot)\) \(\chi_{6762}(3547,\cdot)\) \(\chi_{6762}(3559,\cdot)\) \(\chi_{6762}(4147,\cdot)\) \(\chi_{6762}(4735,\cdot)\) \(\chi_{6762}(5029,\cdot)\) \(\chi_{6762}(5311,\cdot)\) \(\chi_{6762}(5323,\cdot)\) \(\chi_{6762}(5617,\cdot)\) \(\chi_{6762}(5899,\cdot)\)
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
\((2255,3727,3823)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{9}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 6762 }(5899, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{9}{22}\right)\) |
sage: chi.jacobi_sum(n)