from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3864, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,0,55,51]))
pari: [g,chi] = znchar(Mod(1027,3864))
Basic properties
| Modulus: | \(3864\) | |
| Conductor: | \(1288\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1288}(1027,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3864.dw
\(\chi_{3864}(19,\cdot)\) \(\chi_{3864}(283,\cdot)\) \(\chi_{3864}(355,\cdot)\) \(\chi_{3864}(451,\cdot)\) \(\chi_{3864}(523,\cdot)\) \(\chi_{3864}(619,\cdot)\) \(\chi_{3864}(787,\cdot)\) \(\chi_{3864}(1027,\cdot)\) \(\chi_{3864}(1123,\cdot)\) \(\chi_{3864}(1459,\cdot)\) \(\chi_{3864}(1627,\cdot)\) \(\chi_{3864}(1699,\cdot)\) \(\chi_{3864}(2035,\cdot)\) \(\chi_{3864}(2131,\cdot)\) \(\chi_{3864}(2803,\cdot)\) \(\chi_{3864}(3043,\cdot)\) \(\chi_{3864}(3139,\cdot)\) \(\chi_{3864}(3211,\cdot)\) \(\chi_{3864}(3379,\cdot)\) \(\chi_{3864}(3547,\cdot)\)
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
\((967,1933,1289,2761,2857)\) → \((-1,-1,1,e\left(\frac{5}{6}\right),e\left(\frac{17}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3864 }(1027, a) \) | \(-1\) | \(1\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{17}{22}\right)\) |
sage: chi.jacobi_sum(n)