from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3864, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,0,33,11,3]))
pari: [g,chi] = znchar(Mod(143,3864))
Basic properties
| Modulus: | \(3864\) | |
| Conductor: | \(1932\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1932}(143,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3864.dx
\(\chi_{3864}(143,\cdot)\) \(\chi_{3864}(383,\cdot)\) \(\chi_{3864}(479,\cdot)\) \(\chi_{3864}(815,\cdot)\) \(\chi_{3864}(983,\cdot)\) \(\chi_{3864}(1055,\cdot)\) \(\chi_{3864}(1391,\cdot)\) \(\chi_{3864}(1487,\cdot)\) \(\chi_{3864}(2159,\cdot)\) \(\chi_{3864}(2399,\cdot)\) \(\chi_{3864}(2495,\cdot)\) \(\chi_{3864}(2567,\cdot)\) \(\chi_{3864}(2735,\cdot)\) \(\chi_{3864}(2903,\cdot)\) \(\chi_{3864}(3239,\cdot)\) \(\chi_{3864}(3503,\cdot)\) \(\chi_{3864}(3575,\cdot)\) \(\chi_{3864}(3671,\cdot)\) \(\chi_{3864}(3743,\cdot)\) \(\chi_{3864}(3839,\cdot)\)
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
\((967,1933,1289,2761,2857)\) → \((-1,1,-1,e\left(\frac{1}{6}\right),e\left(\frac{1}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3864 }(143, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{6}{11}\right)\) |
sage: chi.jacobi_sum(n)